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University  of  California. 

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Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

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http://www.archive.org/details/firstyearalgebraOOmilnrich 


FIRST  YEAR  ALGEBRA 


BY 


WILLIAM   J.    MILNE,   Ph.D.,  LL.D. 

M 
PRESIDENT  OF  NEW  YORK   STATE   NORMAL  COLLEGE 
ALBANY,    N.Y. 


NEW  YORK    :•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


MS 


MAY  12  1911 
GIFT 

Copyright,  1911,  by 
WILLIAM  J.   MILNE. 

Entered  at  Stationers'  Hall,  London. 


FIRST   TEAR    ALGEBRA 
E.  P.   I 


PREFACE 

This  book  has  been  written  to  meet  the  growing  demand  for 
a  High  School  Algebra  that  contains  only  the  first  year's  work. 
While  the  order  of  topics  resembles  in  general  that  found  in 
the  author's  other  algebras,  yet  a  number  of  changes  have  been 
made,  for  the  purpose  of  simplifying  the  work  and  deferring 
difficulties  until  the  pupil  is  able  to  cope  with  them. 

One  of  the  hardest  ideas  for  the  young  student  to  grasp  is 
that  of  negative  numbers;  and  the  common  practice  of  pre- 
senting them  at  the  very  beginning  of  the  book  results  not  only 
in  the  bewilderment  but  also  in  the  discouragement  of  the  stu- 
dent. In  this  book,  therefore,  the  pupil  is  first  taught  the  sym- 
bols and  the  fundamental  operations  as  applied  to  positive 
numbers,  and  not  until  he  has  become  thoroughly  familiar  with 
these  is  he  introduced  to  negative  numbers.  He  can  thus  con- 
centrate his  entire  attention  on  the  one  new  idea,  and  it  becomes 
a  pleasure  to  him  to  extend  his  knowledge  by  applying  the 
principles  he  has  already  learned  to  the  new  concept.  Again, 
the  troublesome  operation  of  removing  and  inserting  signs  of 
aggregation  is  deferred  until  the  pupil's  gain  in  power  of 
manipulating  algebraic  numbers  renders  the  work  compara- 
tively easy. 

On  the  other  hand,  in  order  to  arouse  from  the  first  the 
interest  of  the  pupil,  simple  problems  to  be  solved  both 
arithmetically  and  algebraically,  as  well  as  easy  solutions  of 
simultaneous  equations  and  of  quadratic  equations  by  factoring, 
are  presented  very  early  in  the  course,  while  the  more  difficult 
phases  of  these  subjects  are  discussed  later.  Throughout  the 
work,  indeed,  the  greatest  emphasis  is  placed  on  equations 
and  problems,  which  furnish  the  most  apt  illustrations  of  the 
practical  uses  of  algebra. 


4  PREFACE 

The  treatment  of  every  principle  is  based  on  the  pupil's 
knowledge  of  arithmetic.  This  close  correlation  of  the  two 
subjects  not  only  illuminates  both  of  them,  but  adds  further 
to  the  simplicity  of  the  book. 

The  problems  are  based  on  interesting  facts  gathered  from 
a  variety  of  sources,  including  physics,  geometry,  and  com- 
mercial life.  A  few  problems  of  the  older  stjde  are  included 
for  the  purpose  of  familiarizing  the  pupil  with  them  and  for 
their  disciplinary  value. 

Graphs  are  presented  in  a  simple  and  comprehensive  man- 
ner, but  the  chapters  are  introduced  in  such  a  way  as  to  render 
practicable  their  omission,  without  disturbing  the  continuity  of 
the  course. 

Factoring  is  thoroughly  taught,  and  the  study  is  greatly 
simplified  by  the  careful  classifying  and  summarizing  of  the 
various  cases. 

New  terms  are  illustrated  or  defined  wherever  they  are 
needed,  the  object  of  this  plan  being  to  prevent  the  confusion 
that  results  in  the  pupil's  mind  from  the  massing  of  large  col- 
lections of  definitions  at  the  beginning  of  each  chapter.  For- 
mal definitions  of  all  terms  are  placed  at  the  end  of  the  book 
in  a  glossary  arranged  in  alphabetical  order. 

Abstract  and  concrete  work  is-  well  balanced,  so  that  the  drills 
in  algebraic  processes  and  representation  are  as  plentiful  as 
the  exercises  for  the  development  of  the  reasoning  faculties. 

Accuracy  is  secured  by  the  numerous  checks,  tests,  and 
verifications  that  are  required  of  the  student,  and  thorough- 
ness is  acquired  through  the  frequent  and  exhaustive  reviews. 

In  the  preparation  of  the  "work,  careful  consideration  has 
been  given  to  the  courses  of  study  outlined  by  the  Regents  of 
the  State  of  New  York  and  by  educational  authorities  else- 
where. The  book  will  be  found  to  meet  the  requirements  of 
these  courses  in  every  particular. 

WILLIAM  J.    MILNE. 


CONTENTS 

PACK 

Intboductiox     ....                          9 

Algebraic  Solutions 12 

Problems 13 

Factors,  Powers,  and  Polynomials 19 

Numerical  Substitution 22 

Review 23 

Fundamental  Operations  (Positive  Numbers)        .         .        .         .26 

Addition 25 

Subtraction 28 

Multiplication 30 

Division .         .34 

Equations  and  Problems 39 

Algebraic  Bepresentation 43 

Pi'oblems 44 

Review 48 

Positive  and  Negative  Numbers 49 

Addition  and  Subtraction 51 

Sum  of  Two  or  More  Numbers 51 

Difference  of  Two  Numbers 54 

Transposition  in  Equations 58 

Algebraic  Bepresentation 61 

Problems 02 

Multiplication 64 

Special  Cases  in  Multiplication 68 

Simultaneous  Equations 74 

Problems 77 

Division 80 

Special  Cases  in  Division 84 

Parentheses .86 

Equations  and  Problems 92 

Literal  Equations 93 

Algebraic  Bepresentation 94 

Problems 96 

Review 98 

5 


C  CONTENTS 

PAGE 

Factoring 101 

Monomials 101 

Binomials 103 

Trinomials 104 

Polynomials 109 

Special  Applications  and  Devices 113 

Review  of  Factoring .  .117 

Equations  solved  by  Factoring 120 

Problems 123 

Fractions .         .         .  125 

Signs  in  Fractions 125 

Reduction  of  Fractions 128 

Addition  and  Subtraction  of  Fractions 135 

Multiplication  of  Fractions 139 

Division  of  Fractions 141 

Complex  Fractions 143 

Equations  and  Problems 146 

Clearing  Equations  of  Fractions 145 

Algebraic  Bepresentation      .        .        .      ■  .        .        .        .  148 

Problems 149 

Review 151 

Simple  Equations .  153 

One  Unknown  Number .153 

Numerical  Equations  .        . 154' 

Literal  Equations 165 

Problems 167 

Solution  of  Formuloe 164 

Simultaneous  Simple  Equations 168 

Two  Unknown  Numbers     .                 168 

Elimination  by  Addition  or  Subtraction      ....  170 

Elimination  by  Comparison 171 

Elimination  by  Substitution 172 

Literal  Simultaneous  Equations 175 

Problems 176 

Three  Unknown  Numbers 182 

Graphic  Solutions 184 

Simple  Equations 184 

Review        .        .^       .        . 195 

Involution 197 

The  Binomial  Formula        '. 200 


CONTENTS  7 

PAGE 

EVOLCTION 203 

Square  Root  of  Arithmetical  Numbers 209 

Roots  by  Factoring 212 

Radicals 213 

Reduction  of  Radicals 218 

Addition  and  Subtraction  of  Radicals 222 

Multiplication  of  Radicals 224 

Division  of  Radicals 226 

Involution  and  Evolution  of  Radicals 227 

Rationalization 229 

Radical  Equations 232 

Review 236 

Quadratic  Equations 237 

•  Pure  Quadratic  Equations 237 

Problems      . 239 

Formidoi 239 

Affected  Quadratic  Equations 241 

Literal  Equations 246 

Radical  Equations 247 

Problems 249 

Formulce 265 

Graphic  Solutions 256 

Quadratic  Equations  —  One  Unknown  Number  ....  256 

Equations  in  Quadratic  Form 260 

Simultaneous  Quadratic  Equations 264 

Problems 272 

Graphic  Solutions 275 

Quadratic  Equations  —  Two  Unknown  Numbers  .275 

Simultaneous  Quadratic  Equations 279 

Ratio  and  Proportion 284 

Ratio 284 

Properties  of  Batios 285 

Proportion 286 

Properties  of  Proportions 287 

Problems 292 

General  Review 296 

Factors  and  Multiples 305 

Highest  Common  Factor 305 

Lowest  Common  Multiple 308 

Glossary 311 


FIRST  YEAR  ALGEBRA 


INTRODUCTION 


1.  In  passing  from  arithmetic  to  algebra,  the  student  will 
not  find  the  change  a  very  marked  one.  He  will  meet  signs, 
definitions,  principles,  and  processes  with  which  he  is  already- 
familiar.  The  fundamental  principles  of  arithmetic  and 
algebra  are  identical,  but  in  algebra  their  application  is 
broader. 

Algebra  uses  the  same  number  symbols  as  arithmetic,  namely, 
1,  2,  3,  4,  5,  etc.,  but  from  time  to  time,  as  need  for  them 
arises,  various  new  symbols  will  be  introduced.  While  arith- 
metic, to  a  limited  extent,  uses  letters  to  represent  numbers, 
their  use  is  a  distinctive  feature  of  algebra. 

The  terms,  addition,  subtraction,  multiplication,  division, 
fraction,  etc. ;  the  associated  terms  such  as  addend,  subtrahend, 
multiplier,  quotient,  numerator;  and  the  signs,  +,  — ,  X,  -^,  =, 
have  the  same  meanings  that  they  have  in  arithmetic ;  but  it 
will  be  seen  that  algebra  gives  to  some  of  them  additional 
meanings. 

In  algebra,  multiplication  is  also  indicated  by  the  dot  (•)  or  by  the 
absence  of  sign  ;  thus,  ax  b,  a  -  b,  and  ab  all  mean  the  same. 

Division  is  often  indicated  by  a  fraction  ;  thus,  o  -*-  6  and  ^  have  the 
same  meaning. 


10  INTRODUCTION 


EXERCISES 

2.   Read, 

and 

tell 

the   meaning  of  each  of 

the  following 

algebraic  expressions : 

1.   2  +  3. 

8.      W  -f-  V.                                     ,  e 

15. 

7>l 

2.   a  +  &. 

- 

9.    4.5. 

71 

3.    8-5. 

10.    x-y,                       16 

a& 

4.    x-y. 

11.  pq. 

3a; 

5.    2x5. 

12.    ab  —  rs.                  17. 

?  +  -• 

6.    m  X  n. 

13.    3v-|-5^. 

a  —  V 

7.    8-4. 

14.   a  +  m-n.             18. 

7^  1   » 

Indicate  the 

19.^  Sum  of  5  and  2 ;  of  a;  and  ?/. 

20.  *  Difference  of  9  and  6 ;  of  m  and  n. 

21.  Product  of  3'  and  4  in  two  ways ;  product  of  r  and  s  in 
three  ways. 

22.  Quotient  of  8  divided  by  5  in  two  ways ;  quotient  of  p 
divided  by  q. 

23.  Sum  of  5  times  d  and  2  times  c. 

24.  Difference  of  a  times  b  and  2  times  4. 

25.  Product  of  3  m  and  n. 

26.  Quotient  oi  v  —  iv  divided  by  c  times  d. 

27.  Product  of  2  a;  +  7  and  3  2/  -  2. 

The  product  of  2  jc  +  7  and  3  y  —  2  is  indicated  thus  : 

(2x  +  7)(3y-2). 
Note.  —  Parentheses,  (  ),  are  used  to  group  numbers,  when  the  num- 
bers in  each  group  are  to  be  considered  as  a  single  number. 

28.  Product  of  a  —  6  and  5  m  +  2. 

29.  Product  of  a  and  a  +  b  divided  by  the  product  of  b  and 
a  —  b. 

30.  A  boy  had  a  apples  and  his  brother  gave  him  b  more. 
How  many  apples  had  he  then  ? 

*In  this  book,  the  '  difference '  of  two  numbers  means  the  first  mentioned 
less  the  second. 


INTRODUCTION  11 

31.  Edith  is  14  years  old.  How  old  was  she  4  years  ago  ? 
a  years  ago  ?     How  old  will  she  be  in  3  years  ?  in  b  years  ? 

32.  At  J?  cents  each,  how  much  will  5  oranges  cost  ? 

33.  If  ijcaps  cost  10  dollars,  how  much  will  1  cap  cost  ? 

34.  At  y  cents  eachXnow  many  pencils  can  be  bought  for  x 
cents  ? 

35.  George  won  a  race  by  running  the  distance  in  t  seconds. 
Kepresent  Elmer's  time,  if  he  took  2  seconds  longer. 

36.  James  weighs  p  pounds.  Represent  Edward's  weight, 
if  he  weighs  10  pounds  less  than  James. 

37.  A  boy  who  had  p, marbles  lost  q  marbles  and  afterward 
bought  r  marbles.     How  many  marbles  did  he  then  have  ? 

38.  If  m  represents  the  number  of  miles  a  boy  can  walk  in 
a  certain  time,  indicate  the  distance  his  father,  who  walks 
twice  as  fast,  can  walk  in  the  same  time. 

39.  Mary  paid  c  cents  for  a  pin  and  half  as  much  for  a  belt. 
Represent  the  number  of  cents  she  paid  for  the  belt.         / 

40.  What  two  whole  numbers  are  nearest  to  9  ?  to  x,  if  x 
is  a  whole  number  ?  to  a,  if  a  is  a  whole  number  ? 

41.  li  y  is  an  even  number,  what  are  the  two  nearest  even 
numbers  ?  the  two  nearest  odd  numbers  ? 

3.,  Unite  terms  as  indicated  by  their  signs  : 

20  2  tens  2x10  2t  2x  '2z 

+  40  +4tens  +4x10  +4«  +4a;  +42 

+  30  +3  tens  +3x10  +3^  +3x  +32 

90  9  tens  9x10  9t 

2^  +  4^  +  3^  =  9^.      2x-\-4:X-]-Sx=?      22  +  42  +  32  =  ? 

Such  terms  as  2  i»,  +  4  a;,  and  +  3  a;  are  called  like,  or  similar, 
terms  because  they  have  the  same  unit,  x. 

The  multipliers,  2,  4,  and  3  are  called  coefficients  of  x. 

Such  terms  as  2  ^,  +  4  a;,  and  3  z  are  unlike,  or  dissinmar, 
terms  because  they  have  different  units,  t,  a;,  and  z. 


12  INTRODUCTION 


ALGEBRAIC   SOLUTIONS 

4.  The  numbers  in  this  chapter  do  not  differ  in  character 
from  the  numbers  with  which  the  student  is  already  familiar 
in  arithmetic. 

The  following  solutions  and  problems,  however,  serve  to 
illustrate  how  the  solution- of  an  arithmetical  problem  may 
often  be  made  easier  and  clearer  by  the  algebraic  method,  in 
which  the  numbers  sought  are  represented  by  letters,  than  by 
the  ordinary  arithmetical  method. 

Letters  that  are  used  for  numbers  are  usually  called  literal 
numbers. 

5.  Illustrative  Problem.  —  A  man  had  400  acres  of  corn  and 
oats.  If  there  were  3  times  as  many  acres  of  corn  as  of  oats, 
how  many  acres  were  there  of  each  ? 

Arithmetical  Solution 
A  certain  number  =  the  number  of  acres  of  oats. 
Then,  3  times  that  number  =  the  number  of  acres  of  corn, 

and  4  times  that  number  =  the  number  of  acres  of  both  ; 

therefore,         4  times  that  number  =  400. 

Hence,  the  number  =  100,  the  number  of  acres  of  oats, 

and  3  times  the  number  =  300,  the  number  of  acres  of  corn. 

Algebraic  Solution 

Let  X  =  the  number  of  acres  of  oats. 

Then,  Sx  =  the  number  of  acres  of  corn, 

and  4  a;  =  the  number  of  acres  of  both  ; 

therefore,  4  a;  =  400. 

Hence,  x  =  100,  the  number  of  acres  of  oats, 

and  3a;  =  300,  the  number  of  acres  of  corn. 

Observe  that  in  the  algebraic  solution  x  is  used  to  stand  for  '  a  certain 
number'  or  'that  number,'  and  thus  the  work  is  abbreviated. 

6.  A  statement  of  the  equality  of  ty^ numbers  or  expres- 
siMs  is  called  an  equation.  J^ 

6  X  =  30  is  an  equation.  *^ 


INTRODUCTION  13 

Problems 
7.   Solve  the  following  problems : 

1.  A  bicycle  and  suit  cost  $54.  How  much  did  each  cost, 
if  the  bicycle  cost  twice  as  much  as  the  suit  ? 

2.  Two  boys  dug  160  clams.  If  one  dug  3  times  as  many 
as  the  other,  how  many  did  each  dig  ? 

3.  Find  a  number  whose  double  equals  52. 

4.  If  3  times  a  number  equals  75,  find  the  number. 

5.  A  certain  number  added  to  3  times  itself  equals  96. 
What  is  the  number  ? 

6.  The  average  length  of  a  fox's  life  is-  twice  that  of  a 
rabbit's.  If  the  sum  of  these  averages  is  21  years,  what  is  the 
average  length  of  a  rabbit's  life  ? 

7.  The  battleship  fleet  that  sailed  for  the  Pacific  consisted 
of  20  ships.  The  number  of  warships  was  4  times  the  num- 
ber of  the  auxiliary  ships.     How  many  warships  were  there  ? 

8.  The  water  and  steam  in  a  boiler  occupied  120  cubic  feet 
of  space,  and  the  water  occupied  twice  as  much  space  as  the 
steam.     How  many  cubic  feet  of  space  did  each  occupy  ? 

9.  One  year  the  United  States  exported  24  million  pounds 
of  butter  and  cheese.  If  this  included  twice  as  much  butter 
as  cheese,  how  many  pounds  of  each  were  exported  ? 

10.  Porto  Rico  and  the  Philippines  together  produce  400,000 
tons  of  sugar  each  year.  If  the  latter  produces  3  timers  as 
much  as  the  former,  how  much  does  Porto  Rico  produce? 

11.  Canada  and  Alaska  together  annually  export  furs  worth 
3  million  dollars.  If  Canada  exports  5  times  as  much  as 
Alaska,  find  the  value  of  Alaska's  export. 

12.  The  poultry  and  dairy  products  of  this  country  amount 
to  520  million  dollata'  a  year,  or  4  times  the  value  of  the 
potato  crop.     What  is  the  value  of  the  potato  crop  ? 


14  INTRODUCTION 

13.  At  Portland,  Oregon,  recently  vessels  were  loaded  with 
25  million  feet  of  lumber  for  home  and  foreign  ports.  Find 
the  foreign  shipment,  if  it  was  4  times  that  to  home  ports. 

14.  In  constructing  the  Galveston  sea  wall  10,000  loads  of 
sand  and  crushed  granite  were  used.  If  there  were  3  times  as 
many  loads  of  sand  as  of  granite,  how  many  loads  of  each  were 
used  ? 

15.  The  Weather  Bureau  of  the  United  States  yearly  saves 
the  country  30  million  dollars,  or  20  times  its  cost.  What  is 
the  annual  cost  of  the  Weather  Bureau  ? 

16.  One  year  in  continental  Europe  6  million  watches  were 
made,  and  this  number  was  i  of  a  million  more  than  twice  the 
number  made  in  the  United  States.  How  many  were  made  in 
this  country  ? 

Suggestion,  2  x  =  6  —  ^. 

17.  Probably  Ceylon  has  the  oldest  tree  in  the  world,  and 
its  age  is  about  2200  years.  If  this  is  70  years  more  than  6 
times  the  age  of  the  Powhatan  Oak  in  Virginia,  find  the  age 
of  the  latter. 

18.  The  value  of  the  King's  Cup,  the  challenge  trophy  for 
yachting,  is  twice  as  great  as  that  of  the  Bennett  Cup,  the  prize 
for  long-distance  balloon  racing.  If  the  difference  in  value 
is  $  2500,  find  the  value  of  each. 

19.  The  owner  of  a  piano  found  that'  the  annual  cost  of 
keeping  it  in  tune  and  insuring  it  against  fire  was  $12.50,  and 
that  the  cost  of  keeping  it  in  tune  was  9  times  the  cost  of 
insupiug  it.     Find  the  cost  of  each  item. 

20.  The  Quebec  bridge  that  collapsed  was  1800  feet  long, 
and  twice  the  length  of  the  Forth  bridge  was  yV  ^f  the  length 
of  that  at  Quebec.     Find  the  length  of  the  Forth  bridge. 

21.  One  year  1500  violins  were  made  in  the  United  States. 
Twice  as  many  were  made  in  New  York  as  in  Massachusetts, 
and  these  two  states  made  half  of  all  that  were  made  in  this 
country.     How  many  violins  were  made  in  New  York  ? 


INTRODUCTION  15 

22.  The  sides  of  any  square  (Fig.  1)  are  equal  in  length. 
How  long  is  one  side  of  a  square,  if  the  perimeter  (distance 
around  it)  is  36  inches  ? 


Fig.  1.  Fig.  2.  Fig.  3. 

23.  The  length  of  each  of  the  sides,  a  and  h,  of  the  triangle 
(Fig.  2)  is  twice  the  length  of  the  side  c.  If  the  perimeter  is 
40  inches,  what  is  the  length  of  each  side  ? 

24.  The  opposite  sides  of  any  rectangle  (Fig.  3)  are  equal. 
If  a  rectangle  is  twice  as  long  as  it  is  wide  and  its  perimeter 
is  48  inches,  how  wide  is  it  ?   how  long  ? 

25.  Divide  21  into  three  parts,  such  that  the  first  is  twice 
the  second,  and  the  second  is  twice  the  third. 

Suggestion.  —  Let  x  =  the  third  part ;  then,  2  x  =  the  second  part, 
and  2  •  2  x  =  the  first  part ;  that  is,  a;  +  2  a;  +  2  .  2 x  =21 . 

26.  Three  newsboys  sold  60  papers.  If  the  first  sold  twice 
as  many  as  the  second,  and  the  third  sold  3  times  as  many  as 
the  second,  how  many  did  each  sell  ? 

27.  The  battleship)  Connecticut  has  twice  as  many  8-inch  as 
12-inch  guns,  and  the  sum  of  the  two  equals  the  number  of  its 
7-inch  guns.  If  it  has  in  all  24  guns  of  these  sizes,  find  the 
number  of  each. 

28.  One  winter  the  Borough  of  Richmond  had  four  falls  of 
snow  amounting  in  all  to  16^  inches.  The  second  and  third 
falls  were  each  4  times  the  first.  Find  the  depth  of  the  fourth 
fall,  if  it  was  twice  the  first. 

29.  In  a  recent  year  Massachusetts  produced  twice  as  many 
barrels  of  cranberries  as  New  Jersey,  and  New  Jersey  6  times 
as  many  as  Wisconsin.  Find  the  production  of  each  of  these 
states,  if  their  total  crop  was  400,000  barrels. 


16  INTRODUCTION 

30  A  plumber  and  two  helpers  together  earned  $  7.50  per 
day.  How  much  did  each  earn  per  day,  if  the  plumber  earned 
4  times  as  much  as  each  helper  ? 

31.  James  bought  a  pony  and  a  saddle  for  $60.  If  the 
saddle  cost  J  as  much  as  the  pony,  find  the  cost  of  each. 

Suggestion.  — Let  x  =  the  number  of  dollars  the  saddle  cost. 

32.  Separate  72  into  two  parts,  one  of  which  shall  be  J  of 
the  other. 

33.  Separate  78  into  two  parts,  one  of  which  shall  be  i  of 
the  other. 

34.  A  skating  rink  accommodated  4000  persons.  If  there 
were  -J  as  many  skaters  as  spectators,  find  the  number  of  each. 

35.  The  total  production  of  sulphur  averages  625,000  tons 
per  year.  How  much  is  produced  by  the  rest  of  the  world,  if 
it  is  I  the  amount  produced  by  Sicily  ? 

36.  The  average  height  of  the  land  above  sea  level  is  -^  as 
great  as  the  average  depth  of  the  ocean,  and  the  sum  of  the 
two  is  13,000  feet.  Find  the  average  height  of  the  land  and 
the  average  depth  of  the  ocean. 

37.  The  first  issue  of  Christmas  stamps  by  the  Delaware 
Red  Cross  Society  was  ^  as  much  as  the  second,  which  was  i 
as  much  as  the  third.  If  the  three  issues  amounted  to  350,000 
stamps,  how  many  were  there  in  each  issue  ? 

38.  Sand  and  clay  road  costs  ^  as  much  per  mile  as  macadam. 
If  the  former  costs  $  400  per  mile,  find  the  cost  of  the  latter. 

Solution 

Let  X  =  the  number  of  dollars  macadam  costs  per  mile. 

Then, '  ^  x  =  400. 

Therefore,  x  =  6  times  400  =  2400. 

Hence,  macadam  road  costs  $  2400  per  mile. 


39.   The  gold  output  of  the  United  States  for  a  recent  year 
as  110  million  dollars,  or  \  that  of  i 
was  the  world's  output  for  that  year  ? 


was  110  million  dollars,  or  \  that  of  the  entire  world.     What 


INTRODUCTION  17 

40.  A  man  in  New  York  rented  Ms  house  and  lived  in  an 
apartment  costing  him  $  2000  a  year.  This  was  ^  as  much  as 
the  rent  of  his  house.     For  how  much  did  his  house  rent  ? 

41.  The  Pennsylvania  Railroad  station  in  New  York  is  780 
feet  long,  and  this  is  404^  feet  more  than  ^  the  length  of  the 
Capitol  at  Washington.     Find  the  length  of  the  Capitol. 

42.  A  basketball  team  won  16  games,  or  J  of  the  games  it 
played.     Find  the  number  of  games  it  played. 

Solution 

Let  X  =  the  number  of  games  it  played. 

Then,  I  x  =  16, 

and  ^x  =  S. 

Therefore,  x  =  24,  the  number  of  games  it  played. 

43.  The  largest  thermometer  in  the  world  has  a  glass  tube 
16  feet  loug.  Find  the  length  of  the  thermometer,  if  the  tube 
is  ^  of  the  entire  length. 

44.  What  is  the  annual  rainfall  of  Hawaii,  if  at  least  56 
inches,  or  ^  of  it,  passes  off  without  rendering  any  service  ? 

45.  Of  the  inhabitants  of  Guam,  -3%,  or  8100,  can  read  and 
write.     What  is  the  population  of  the  island  ? 

46.  The  average  annual  fire  loss  in  Berlin  is  ^  of  that  in 
Chicago.  If  the  fire  loss  in  Berlin  is  $150,000,  what  is  the 
fire  loss  in  Chicago  ? 

47.  The  largest  stone  ever  quarried  in  the  South  was  dressed 
down  to  weigh  60,000  pounds.  If  this  was  |  of  its  weight  as 
originally  blocked  out,  find  its  original  weight. 

48.  Find  the  amount  of  lumber  on  hand  in  San  Francisco  at 
the  time  of  the  earthquake,  if  |  of  it,  or  36  million  feet,  were 
consumed  by  the  fire  that  followed  the  earthquake. 

49.  The  manufacturing  industries  of  Great  Britain  use 
150  million  tons  of  coal  per  year.  If  this  is  f  of  the  total 
amount  used,  what  is  that  country's  annual  consumption  of 
coal  ? 

MILNE's   IST  YR.   ALG.  2 


18  INTRODUCTION 

50.  The  number  of  German-speaking  people  in  the  world  is 
75  million,  or  f  the  number  that  speak  English.  What  is  the 
number  of  English-speaking  people  ? 

51.  The  United  States  sent  to  Germany  one  year  135,000 
pairs  of  shoes.  This  was  f  of  the  number  sent  the  next  year. 
How  many  pairs  of  shoes  were  sent  the  second  year  ? 

62.  If  I  of  a  number  is  added  to  the  number,  the  sum  is  12. 
What  is  the  number  ? 

Suggestion.  x  -\-  ^x  =  12  ;  that  is,  |  a;  =  12. 

53. ,  If  ^  of  a  number  is  subtracted  from  twice  the  number, 
the  difference  is  35.     What  is  the  number  ? 

Suggestion.  2  a;  —  ^  x  =  35  ;  that  is,  |  a:  =  35, 

54.  The  total  cost  of  the  Pennsylvania  Capitol  was 
13  million  dollars.  If  the  furnishings  cost  2i  times  as  much  as 
the  construction,  what  was  the  cost  of  each? 

55.  The  retail  dressmaking  trade  each  day  uses  -J  of  the 
total  daily  output  of  spool  silk.  If  the  manufacturing  trade 
uses  the  remainder,  or  16,000  miles,  how  much  does  the  dress- 
making trade  use  per  day  ? 

56.  Out  of  the  average  daily  output  of  stamped  envelopes 
^  are  plain  stamped.  The  remainder,  2,800,000,  bear  the 
return  address.     What  is  the  daily  output  ? 

57.  In  one  year,  5600  tons  of  dynamite  were  required  for  the 
Panama  Canal.  If  the  amount  for  the  Culebra  division  was 
If  as  much  as  that  for  the  rest  of  the  canal,  find  the  amount 
required  for  the  Culebra  division. 

58.  In  the  first  twenty-one  hours  after  the  institution  of 
regular  wireless  service,  6|-  times  as  many  words  were  sent  to 
Europe  as  were  received,  and  the  number  sent  was  11,000 
more  than  the  number  received.  Find  the  number  sent; 
the  number  received. 

59.  The  Pacific  battleship  fleet  carried  twice  as  much  ham 
as  it  did  salt  pork,  and  2^  times  as  much  beef  as  it  did  ham. 
The  weight  of  the  beef  was  800,000  pounds  more  than  that  of 
the  salt  pork.     Find  the  weight  of  each. 


INTRODUCTION  19 


FACTORS,  POWERS,  AND  POLYNOMIALS 

8.  Since  the  product  of  2  and  6  or  of  3  and  4  is  12,  each  of 
the  numbers  2,  6, 3,  and  4  is  a  factor  of  12.  So  also,  each  of  the 
numbers  3,  a,  b,  3  a,  3  b,  and  ab  is  a  factor  of  ^  a6. 

9.  In  algebra,  as  in  arithmetic,  such  a  product  as  2  x  2  x  2  x  2, 
called  a  power  of  2,  may  be  more  briefly  written  2*. 

The  small  figure  4,  placed  at  the  right  of,  and  a  little  above, 
the  2  to  indicate  the  number  of  times  2  is  used  as  a  factor,  is 
called  an  exponent. 

Since  a^  means  the  same  as  a,  the  exponent  1  is  usually  omitted. 
a^  is  read  '  a  second  power '  or  '  a  square  ' ;  a^  is  read  '  a  third  power ' 
or  *  a  cube ' ;  a^  is  read  '  a  fourth  power,'  '  a  fourth,'  or  '  a  exponent  4.' 

The  terms  ^coefficient'  and  ^exponent'  should  be  distin- 
guished. 

Thus,  5  a  means  a-ha-{-a-\-a-\-a,  but  a^  means  a    a  •  a-  a-  a. 

EXERCISES 

10.  Kead,  and  tell  the  meaning  of : 

1.  a-«.  4.   Q^y\  7.   3  2wl  iq.   9a6Vd*. 

2.  y^.  5.    a^b\  8.   4/)Y-  H-   ^pVs't^. 

3.  2*.  6.    r's-.  9.    2mV.  12.    7  x^ym^n*. 

Express  in  abbreviated  form  by  using  exponents : 

13.  2  .  2.  16.   3  aaa.        19.    2  .  2  .  2  .  a;  .  «. 

14.  3-3.3.  17.   8  nil  20.    7  '7  'Z  .z  'Z  -z. 

15.  5  •  5  •  5  •  5.       18.   9  ssrrr.       21.    SS'Sa-a-bbb. 

22.  What  is  the  coefficient  of  a;  in  3  a; ?  in  ax?  in  3 aa; ? 

Note. — A  coefficient  is  numerical,  literal,  or  mixed  according  as  it  is 
composed  of  figures,  letters,  or  both. 

When  not  otherwise  specified  '  coefficient '  means  numerical  coefficient. 
Since  la  means  the  same  as  a,  the  coefficient  1  is  usually  omitted. 

23.  What  is  the  literal  coefficient  of  f  in  at-  ?  in  gt^  ?  in 
nH^  ?     of  2^  in  nf  ?     in  arf  ?     in  bnf  ? 


20  INTRODUCTION 

Name  the  various  factors  of  : 

24.  ax.  26.    a^.  28.    6n.  30.  pqrs. 

25.  Smn.  2t.   5r^^.  29.  15  z\  31.    24  v«. 

32.  In  each  of  the  exercises  24-31,  name  the  factors  in  sets 
such  that  the  product  of  the  factors  in  each  set  shall  equal  the 
given  number. 

11.  An  algebraic  expression  is  called  a  monomial,  binomial, 
or  trinomial  according  as  it  has  one,  two,  or  three  terms. 

Thus,  3  a  is  a  monomial ;  2  ic  +  y^,  a  binomial ;  and  x^  +  2xy  ■{■  y'^,  a 
trinomial. 

The  name  polynomial  is  often  applied  to  any  algebraic 
expression  of  more  than  one  term. 

EXERCISES 

12.  From  the  algebraic  expressions  given  below  select  the : 

1.  Binomials.  3.    Monomials.         5.    Similar  terms. 

2.  Trinomials.         4.    Polynomials.      6.    Dissimilar  terms. 

2ax',  Sa^y,  2a4-36;  3a;  +  26;  Sax-{-2y^; 

6a  —  c-\-d;  3  a^x^  —  4:ax-\-2d  —  y^;  2x^y  —  xy  +  a  V. 

7.  Find  the  value  of  3  +  4-2  +  3;  of  3x4^2x3. 

Solutions.      3+4-2  +  3  =  7-2  +  3  =  5  +  3  =  8; 
3x4-2x3  =  12 -2x3  =  6x3  =  18. 

When  only  +  and  —  occur  in  any  expression^  or  only  x  and  -4-,  the 
operations  are  performed  in  order  from  left  to  right. 

Find  the  value  of : 

8.  3-2-1  +  8-3  +  4.         10.    10--2x8--4--2x6. 

9.  5_j_i_4_|.3_2  +  6.         11.    35-f-7--5x3x4--2. 

12.    Find  the  value  of  7  + 10  -  6  ^  3  X  4. 

Solution.     7  + 10 -6-3x4  =  7  + 10 -2x4  =  7  + 10 -8  =  9. 

Unless  otherwise  indicated,  as  by  the  use  of  parentheses,  when  x,  -^, 
or  both,  occur  in  connection  with  +,  — ,  or  both,  the  indicated  WiUUi^tv^a- 
tions  and  divisions  are  performed  first. 


« 


INTRODUCTION  21 

Find  the  value  of : 

13.  5x10-7.  18.  6  +  2x8-4h-2. 

14.  5x(10-7).  19.  (6  +  2)  X  8-4-2. 

15.  2x5  +  3x4.  20.  (6  +  2x8-4)^-2. 

16.  (25-13)-4x2.  21.  6  +  2x(8-4)-2. 

17.  16-2x2xl2-f-4.  22.  6  +  2  x(8h-4 --2). 
Kead,  and  tell  the  meaning  of  each  of  these  polynomials : 

23.  2x^  +  y\       26.    a-hd{ax-y).  29.  Sa^-\-2y-3z. 

24.  3x-4.y.      27.   3-\-A(y-3z).  30.  a^x- - 3 xy  +  2 z-. 

25.  4a6-c3.       28.    c(P-{-e)-^d.  31.  5  6"?/ +  ary  +  5 c;^^ 
Represent  algebraically : 

32.  The  sum  of  five  times  a  and  three  times  the  square  of  x. 

33.  Three  times  b  less  twice  the  fifth  power  of  a. 

34.  The  product  of  a,  b,  and  a  —  c. 

35.  Three  times  x,  divided  by  five  times  the  sum  of  a,  6, 
and  c. 

36.  Seven  times  the  product  of  x  and  y,  increased  by  three 
times  the  cube  of  z. 

37.  Six  times  the  square  of  m,  increased  by  the  product  of 
m  and  n. 

38.  The  product  of  a  used  five  times  as  a  factor,  multiplied 
by  the  sum  of  6  and  c. 

39.  Twelve  times  the  square  of  a,  diminished  by  five  times 
the  cube  of  b. 

40.  Eight  times  the  product  of  a  and  b,  divided  by  four 
times  the  seventh  power  of  c. 

41.  Six  times  the  product  of  a  second  power  and  n,  increased 
by  five  times  the  product  of  a  and  the  second  power  of  7i. 

42.  The  fourth  power  of  the  sum  of  a  and  b,  increased  by 
three  times  the  product  of  the  square  of  a  and  the  square  of  b, 
diminished  by  the  cube  of  d. 


22  INTRODUCTION 


NUMERICAL  SUBSTITUTION 

13.  When  a  particular  number  takes  the  place  of  a  letter,  or 
general  number,  the  process  is  called  substitution. 

EXERCISES 

14.  1.  When  a  =  2  and  6  =  3,  find  the  numerical  value  of 
Sab;  of  a*. 

Solutions.     3 a&  =  3  •  2  •  3  =  18  ;  also,  a*  =2  ■  2  ■  2  •  2  =  16. 

When  a  =  5,  6  =  3,  c  =  10,  m  =  4,  find  the  value  of : 

2.  10  a.  6.   5  m".  10.    am\  14.   ^ab\ 

3.  2ab.  7.   2a^b.  11.    (abf.  15.   ^bm. 

4.  3  cm.  8.   3  6m^  12.    a^b\  16.   i-abc, 

5.  6  be.  9.    4a^6.  13.    a^c.  17.   362cml 

18.  When  m  =  0  and  n  =  4,  find  the  value  of  3  m^n. 
Solution.  3  m^n  =  3  .  O^  .  4  =  3  •  0  •  4  =  0. 

Note.  —  When  any  factor  of  a  product  is  0,  the  product  is  0;  there- 
fore, any  power  of  0  is  0. 

When  a  =  4,  6  =  2,  r  =  0,  and  s  =  5,  find  the  value  of : 

19.  7  6V.  21.    3s''6''.  23.    fa»6s.         25.    2«6VV1 

20     i^  22     'Jl^  24     ^^.  26     ^^  ^'^'^° 

'     s6   *  '   abs'-'  '    b'^a'  '    Ga'b's' 

27.  When  x  =  S  and  ?/  =  2,  find  the  value  of  (a5  +  2/)^;  of 
x'^2xy  +  f. 

Solutions 

(x  +  yy  =  (3  +  2)2  =  5  .  5  =  25. 
.r2  +  2  x?/  +  2/2  =  3  .  3  +  2  .  3  .  2  +  2  .  2  =  9  +  12  +  4  =  25. 

28.  Show  that  2x-\-Sx  =  5x  when  a;  =  2;  when  a;  =  3. 
Giving  X  any  value  you  choose,  find  whether  2x-{-Sx  =  5x. 

29.  Show  that  m  (a  -f  6)  =  ma  +  m6  when  m  =  5,  a  =  4,  and 
6  =  3.  Find  whether  the  same  relation  holds  true  for  other 
values  of  m,  a,  and  6. 


REVIEW  23 

30.   Showthat  (a-6)2  =  a--2a6  4-&^wheua  =  4  and  6  =  2. 
Find  whether  this  is  true  for  other  values  of  a  and  b. 

When  a  =  5,  6  =  3,  m  =  4,  ?i  =  1,  find  the  value  of : 


31.    a'  +  b\ 

33.    71^-1.                   35.    m»-^ 

32.    (a  +  bf. 

34.    (n-iy.                 36.    (6??i)*- 

37.    a6  —  bn  -\-  mb^  -i-  3  mn^. 

38.    (a6  —  6/1  4-  m6^)  -5-  3  m?i^. 

39.  2"wi^w-  —  abmn  -^  4  &?i  —  ??i^ri^. 

40.  a^nbn^  —  1 6^??^  +  |  m V  —  |  m^. 

REVIEW 

15.  Bead  the  following ;  classify  each  expression  according 
to  the  number  of  terms  it  contains ;  find  the  number  repre- 
sented by  each  expression  when  t  =  10. 

1.  6^.  4.   f.  7.    t^. 

2.  7t.  5.    ^2  +  2^  +  1.  8.    t^-\-2t--{-5t  +  4:. 

3.  9^  +  9.  6.    3^2^6^4-5.  9.   5f-^St^+ St-{-6. 

10.  Write  25  as  a  polynomial  in  t^  t  representing  10 ;  letting 
t  represent  10,  and,  using  exponents  to  represent  powers  of  t, 
express  in  polynomial  form  : 

732  523  893  4867  6248  12,mn 

11.  What  does  2  a  denote  ?  a^? 

Illustrate  the  difference  in  meaning  between  2  a  and  a^  when 
a  =  1 ;  when  a  =  2 ;  when  «  =  3  ;  when  a  =  ^  ;  when  a  =  J. 
For  what  value  of  a  are  2  a  and  a^  equal  ? 

12.  Which  is  the  greater,  2^  or  3^?  4^  or  2*?  2«  or  5^? 

13.  Compare  also  2«  and  2^;  (if  and  (|)2;  1^  and  1^ 

14.  Find,  for  x  =  1,  the  value  of : 

3iB         4if2  6ar^  8;r'- 4.T^  +  2ar^-;v-h  5 

Name  the  exponent  of  x  in  each  term  that  contains  x. 

15.  Name  the  coefficient  of  n  in  each  of  these  monomials : 
2  w  It  \  n  bn  3  b''n  a-b^n 


24  REVIEW 

16.  Write  three  similar  monomials;  four  dissimilar  mo- 
nomials. 

17.  If  71  is  a  whole  number  greater  than  1  and  a  is  any  num- 
ber, what  is  the  meaning  of  a"  ? 

Find  the  value  of  each  of  the  following  expressions  when 
a  =  5,  &  =  4,  c  =  3,  d  =  2,  e  =1,  and  n  =  3. 

18.  6ab;  2cd;  4w&d;  ^ea;  ndJ'+K 

19.  Sd'h)  3ab';  3{ahf',  d'n^;  (diif-,  d''-^ 

20.  a  +  b^d  —  n-^e.  22.    10-^d  +  S^n  —  e. 

21.  a(b  —  d)-\-a  —  7i^c.  23.    10  ^  (d -\- 3) -\- ac  ^  n. 

24.  c*+c^  +  2c^-2c2-3c  +  3. 

25.  d^-hd«-+-3d«-5(i*  +  2d3-4c?2  +  8c?-l. 

26.  For  what  value  of  a;  is  12  a;  equal  to  72  ? 

Write  ^  12  07  is  equal  to  72 '  as  an  equation.  Solve  the 
equation. 

Express  in  algebraic  form  ;  solve  equations  when  you  can  : 

27.  Three  times  a  certain  number,  x,  is  21. 

28.  The  sum  of  a  certain  number  and  three  times  the  num- 
ber is  40. 

29.  Six  times  a  number,  less  4  times  that  number,  is  13. 

30.  The  distance  around  a  square  lot,  each  side  a  feet  long, 
is  1280  feet. 

31.  Half  of  a  certain  number  is  17. 

32.  Twice  a  certain  number,  less  J  of  the  number,  equals  15. 

33.  Mary  had  m  books  and  James  had  twice  as  many,  the 
two  together  having  18  books. 

34.  John  had  50  cents,  spent  c  cents,  and  earned  d  cents. 
How  much  money  had  he  then? 

35.  I  bought  2  bottles  of  olives  at  b  cents  per  bottle,  3 
packages  of  crackers  at  p  cents  per  package,  and  a  small  cheese 
for  c  cents.  How  much  did  I  expend  for  all?  How  much 
money  had  I  left  out  of  a  dollar  ? 


FUNDAMENTAL    OPERATIONS 


16.  In  this  chapter  the  student  will  use  numbers  he  has 
used  in  arithmetic  and  letters  to  represent  such  numbers.  He 
will  notice  that  the  processes  of  addition,  subtraction,  multipli- 
cation, and  division  here  are  performed  as  in  arithmetic. 

ADDITION 

17.  To  add  monomials. 

1.  How  many  are  2  plus  5?  How  mauy  times  a  number 
are  2  times  the  number  plus  5  times  the  number  ? 

2.  If  n  stands  for  a  number,  how  many  times  n  are  2  times 
n  plus  5  times  w  ?    2  ?i  +  5  n  =  ? 

3.  2x-\-6x=z?  4.   2r  +  5r=?  5.   2«  +  6«  =  ? 

6.  How  many  are  3  4-  4  -f  6  ? 

7.  How  many  days  are  3  days  +  4  days  +  6  days  ? 

8.  3d  +  4d  +  6d  =  ?  9.    3y  +  4y  +  6y=? 

EXERCISES 

18.  1.    Add  4  a  and  3  a. 

PROCESS         Explanation.  —  Just  as  3  a's  and  4  a's  are  7  a's,  so  3  a 
A  +4a  =  7a;  that  is,  when  the  monomials  are  similar  the  sum 


3a 
la 


may  be  obtained  by  adding  the  numerical  coefficients  and 
annexing  to  their  sura  the  common  literal  part. 


Add: 

2.    3  3.   3  a;            4.    7             5.    7  m             6.   3y 

6  6a;                 1                     m                  4y 

""  "26 


26  FUNDAMENTAL   OPERATIONS 


Add: 

7.    2n 
5n 

8.   3  a; 

8a; 

9.   4:xy 

1  xy 

10.  3mw2 
9mn^ 

11.    5r 
2r 
4r 

12.   9^i 
4n 
6n 

13.    2a6 
4a6 

14.  Gc^d^ 
8  c^c?-^ 

Perform  the  additions  indicated : 

15.  8a-h2a  +  a  +  3a+a  +  7c(. 

16.  52/-|-32/  +  82/  +  10.y  +  6^  +  y  +  22/. 

17.  8  m  +  3  m  +  5  m  +  2  m  +  6  7;i  4-  4  m. 

18.  7  6c  +  6c  +  4  6c  +  5  6c  +  8  6c'  +  3  6c. 

19.  4  a;22/2  ^  5  ^^^2  _^  3  ^.y  _,_  ^^2  _^  ^0  x^?/-'  +  6  a;^^^^^ 

20.  3(a6)2  +  9(a6)2  +  (a6)2  +  7(a6)2  +  9(a6)2  +  2(a6)2  +  {abf. 

21.  5(a;+.v)+2(a;+?/)H-3(a;+2/)+8(a;  +  2/)+2(a;+.y)  +  (a;+2/). 

22.  4(a  +  6)^  +  11  (a  4-  6)^  +  7(a  +  6)^  +  2(a  +  6)^  +  5(a  +  6)^. 

Only  similar  terms  can  be  united  into  a  single  term.  Dis- 
similar terms  are  considered  to  have  been  added  when  the 
addition  is  indicated. 

23.  Add  6  a,  5  6,  2  a,  3  6,  2  c,  and  a. 

Solution.  —  Sum  =6a  +  2a+a  +  56  +  a6  +  2c  =  9a  +  86+2c. 

Add : 

24.  2  a;,  4  a,  3  a;,  and  a.  27.  5  r,  f  t,  2  r,  and  \  t. 

25.  m,  3  c,  6  m,  and  4  c.  28.  ^p,  f  Q',  i  p,  and  |  Q'. 

26.  4  w,  -u,  3  w,  and  10  v.  29.  c?,  .4  6,  .5  d,  and  .6  6. 

30.  2  m,  nm,  n,  2  m^i,  3  7n,  4  n,  and  5  mn. 

31.  3  6,  2  a,  2  6,  2  c,  2  d,  0,  c,  6,  4  d,  and  3  c. 

32.  rs,  3  r-s,  4  rs^  2  ?-s,  rs^,  4  r%  2  rs^  and  3  rs. 

33.  3  a;.v,  2  pg,  7  cd,  j9g,  2  cd,  8  pg,  4  cd,  and  2  a;?/. 

34.  a;2,  4  xy,  7  2/^  2  xy,  3  2/',  6  a;^,  /,  xy,  5  a;2,  and  4  /. 


FUNDAMENTAL  OPERATIONS  27 

19.   To  add  polynomials. 

EXERCISES 

1.  Add  X  -^  2  y  -}-  3  z,  x-\-y,  and  x-\-^y  -\-z. 

PROCESS 

Explanation.  —  For  convenience,  similar  terms 
x-^  ^  y  +  6  z       jj^g^y  ^jg  written  in  the  same  column. 
^-\-     y  The  sum  of  the  first  column  is  3  x,  of  the  second 

x  +  iiy  -\-     z      Ty-,  of  the  third  4  z  ;  the  sum  of  these  dissimilar 
3  a;  4-  7  w  4-  4  2      terms  is  then  indicated. 

Add: 

2.  2a4-46  3.   4r+3s  4.    .r'-h2a«/-f^ 
6a-^2b                          r-h    s  x^  +y- 

a+36  3r-\-2s  Sxy  +  y^ 

5.  Add  2  c  +  5  d,  7  c  -h  d,  d  -f  4  c,  and  2  d  +  c. 

6.  Add  6m4-4n,  2m-h3n,  5n-f7m,  and  2  n  -h  3  m. 

7.  Add  ab  +  a^c+5,  3  a6  +  3  a2c  +  7,  and  2a2c-|-2  a6 +  3. 
Express  in  simplest  form: 

8.  2a  +  26-|-3c-f 46  +  4a-|-6a4-2c. 

9.  Sw-\-4:X-{-7  y-{-2v  +  2tv  +  x-^3y-^4:V+3x  +  4:W  +  v. 

10.  a^z  -{-  5  xz^  -\-7  xy  -\-  6  xz^  +  2  x^z  -\-  4:xy  -\-  4  x^z  -^  xz^  -\-  xy. 
Add: 

11.  6  m  +  8  n  4-  X*  +  2/,  2  ?>i  +  2  ?i  +  3  a;  +  4 1/,  and  m  +  x-\-y. 

12.  3  a;  -h  7  2/  4-  4  «4-  6  w,  7  z  4-4  a;  -f  2  y-hw,  and  a;4-t/4-2;  4-w. 

13.  a^  +  2xy-{-y^  2x'+xy-\-y',  ^  ^  xy  +  y\  3xy  +  y''  +  x', 
2o?-\-3xy  +  'if,x'-\-xy  +  2f,2in^2xy  +  3x'  +4:f. 

14.  2c+7d4-6n,  Ilm4-3c4-5ri,  ln-\-2d+%c,  d4-3m4-10c, 
4  fZ  4-  3  ?i  -f  8  m,  m-\-^n,  and  2  m  4-  3  cf . 

15.  3  a;"»  4-  2  y"",  4  a;"*  4-  5  y",  2  a;*"  4-  7  iT,  and  2  x^  4-  y". 

16.  42/-4-2«  +  t(A  y^-{'2w^-\-3z^,  5^'4-3^/;^  2y«4-w;*,  and 


28 


FUNDAMENTAL  OPERATIONS 


SUBTRACTION 

20.    1.    How  many  are  8  less  3  ?     How  many  times  a  num- 
ber are  8  times  the  number  less  3  times  the  number  ? 

2.  Letting  n  stand  for  a  number,  how  many  times  n  are  8 
times  n  less  3  times  n?     Sn  —  3n  =  ? 

3.  8z-3z  =  ?  4.    8s-3s=?  5.    Sa-3a  =  ? 


EXERCISES 

21.    1.    From  10  a  subtract  4  a. 

Explanation. — Just  as  10  a's  less  4  a's  are  6  a's,  so 

10  a  10  a  —  4a=6a;    that  is,  when   terms  are  similar  their 

4  ^  difference  may  be  obtained  by  subtracting  the  numerical 


6  a 


coefficients  and  annexing  the  common  literal  part. 


2.            3. 

4. 

5. 

6. 

7. 

From 

9           9x 

7 

1  ah 

18  m^ 

20  xy 

Take 

4           4x 

3 

Sab 

13  m^ 

16  xy 

8. 

9. 

10. 

11. 

From 

16  ax" 

14^^53 

8  xYz 

21  (a +  6) 

Take 

9ax^ 

Tr'^ 

6  x^yh 

11  (a +  6) 

12. 

13. 

14. 

15. 

From 

3i)-f8r/ 

4?  +  2« 

9x  +  ly 

r 

5r4-8s 

Take 

2p  +  4g 

41+    t 

2x  +  3y 

r 

2r  +  5.«* 

16. 

17. 

18. 

From 

n-^5n'-h2 

'n^          3r4-2s  +  i 

8a2  +  2a&+36=^ 

Take 

n+     n^-\- 

n^              1 

'+     s-\-t 

5a2-f 

a6  +  2  62 

19.  From  12x  +  l  y  subtract  8  a;  +  3  2/. 

20.  From  10  a&  -|-  3  c  subtract  5ah  -\-2c. 

21.  From  7r-\-5s-\-6t  subtract  3r4-2s  +  5^. 

22.  From  9  a^  +  8  2/^  -f  6  a;?/  subtract  5  a;^  +  3  2/^  +  2  a??/. 


FUNDAMENTAL   OPERATIONS  29 

23.  From  5mH-7n4-8Z-f-6  subtract  5m  +  4n  +  4Z4-5. 

24.  From  7  ar^+  32/  +  62;  +  4v  subtract  3a^  +  22/-f50  +  3v. 

Subtract : 

25.  2a:3_^^_^3^from  4a^^72/»  +  5r3. 

26.  4a6  +  262  +  2cdfrom6a6  +  362  +  6cd. 

27.  3a:2v_|.a^^5  from  9a:22^-f6x^H-a^  +  8. 

28.  2  ?;%^  4-  2  vw  +  4  i^^  from  12  -v^  +  9  v^w;^  +6  vw. 

29.  5  m2?ia^  +  abd  from  18  m^na^  +  12  a^feV  +  4  abd. 

30.  4  a;*"  +  2  rC^y"  +  5  2/*'  from  7  a;"*  +  2  oTy''  +  9  y"- 

31.  6  m'  -f- 11  m'n*  4-  5  n'  from  10  m'  -h  1 1  m'n'  -f  8  w'. 

32.  a"*^"  -h  3  6'"-"  +  7  c^"  from  3  a"'^-"  +  5  ft'"""  +  9  c^". 

33.  10  (m  +  n^)  -f-  5  (m^  -\-  n)  from  12  (m  +  7i-)  +  8  (m^  +  n). 

Simplify,  adding  or  subtracting  in  order  as  signs  indicate : 

'  34.  9a;-4xH-6a;.  39.   8  r- 6r  +  5  r-2  r. 

35.  5n  +  3n— 7ri.  40.   7y-\-Sy—6y-^7y. 

36.  8a  — 6a  — 3a  Al.   5z-\-7z  —  2z  —  4:Z. 

37.  2s  +  8s  — 5s.  42.  9v  — 3v  +  2v  — 5u 

38.  3  6— 26  +  7  6.  43.  7n  —  2w  — 3 w  +  4 w. 

44.  8a;  +  7a;-3a:  +  4a;  — 2«-3a;4-6a:. 

45.  2?/  +  3?/-2/  +  7?/-32/  +  92/  +  22/-6?/. 

46.  9z-5z-\-(jz-3z-\-4.z  +  2z-7z  +  Sz. 

47.  5v-{-6v-\-2v  —  5v-\'4:V  —  6v-{-9v  —  ^v  —  Sv. 

48.  7  m  +  6  71  —  3  m  +  5  ?i  4-  7  m  —  4  »i  +  3  wi  +  4  ?i  +  5  m. 

49.  9r+8s+7r-2s-f9s  —  3rH-2r— 7s  +  6s— 5  r+4  r. 

50.  2Z-|-9i+3Z-/4-3«+2<+8Z-5«-f-9Z-6Z+2^-4«+7f. 

51.  10 (a  —  X) 4-  lo(a  —  x)-\-  7 (a  —  x)—  lS{a  —  x)  —  12(a  —  x). 


30  FUNDAMENTAL  OPERATIONS 

MULTIPLICATION 

22.  Product  of  two  monomials. 

In  algebra,  as  in  arithmetic,  the  product  of  two  numbers 
contains  all  the  factors  of  both  numbers,  arranged  or  grouped 
in  any  way  we  please. 

Then,  since  o?  =  aa  and  a^  =  aaa, 

or  '  a^  =  (aa)  (aaa)  =  aaaaa  =  a^. 

That  is,  a^ '  a^  =  a^+^  =  a^.  (Add  exponents) 

Similarly,    3  a^  •  5  a^  =  (3  •  5)  (a'  •  a^)  =  15  a\ 

(Multiply  coefficients) 

Again,      3  a=^6  •  5  a'b'  =  (3  •  5)  (a'  -  a')  (6^  •  b')  =  15  a'b^ 

Hence,  for  multiplication : 

23.  Law  of  exponents.  —  TJie  exponent  of  a  number  in  the 
product  is  equal  to  the  sum  of  its  exponents  in  multiplicand  and 
multiplier. 

24.  Law  of  coefficients.  —  The  coefficient  of  the  product  is  equal 
to  the  product  of  the  coefficients  of  multiplicand  arid  multiplier. 

EXERCISES 

25.  Tell  products  quickly: 

1.      7  a  2.   3i»  3.     5  m  4.   Sab^x^ 

3  a  4  iy  2  m^n  b*cx* 


21  a^ 

12  xy 

10  m'n 

8  ab^cx^ 

5. 

3,7 

6.    4a;2 

7. 

Sav 

8. 

12  a^bc 

4y 

7fl^ 

3  aw 

Sa'b'd' 

9. 

ab' 

10.    3xf 

11. 

2  ax 

12. 

16  c'd^m 

a'b 

9xz 

2  by 

2(^d'n 

13. 

xSj 

14.   7^97^ 

15. 

Sc'd 

16. 

2axft 

xy 

p'q 

4rdh 

6  a^yzh 

FUNDAMENTAL  0PP:RATI0NS  31 

26.   To  multiply  a  polynomial  by  a  monomial. 

Multiplying  as  in  arithmetic,  we  have : 


1. 

43 

2 

86 

321 

3 

963 

40+3 
2 

4  tens  +  3  units 

2 

4^  +  3?/ 
2 

2. 

80  +  6 

300  +  20  +  1 

3 

900  +  60  +  3 

8  tens  +  6  units 
3 

x  +  y  +  z 
a 

ax  +  ay-^  az 

27.  The  product  of  a  polynomial  by  a  monomial  is  equal  to  the 
sum  of  the  partial  products  obtained  by  multiplying  each  term  of 
the  polynomial  by  the  monomial. 


EXERCISES 


28. 

1. 

Multiply : 

a^  +  2                    2.  ax^-\-y 

4                               ax 

4x^-^ 
Sx" 

3.   5mh-\-2t 
3  St' 

4. 

m^  +  n^                 5.  x^-{-2xy-\-y- 
mn                            xy 

6.    xy-\-yz-\-xz 
xyz 

7. 

l4.2.x  +  6ar^  +  4a^                      8. 
x' 

^6a:2  +  2ic  +  l 

In  exercise  7,  the  multiplicand  is  arranged  according  to  the 
ascending  powers  of  x-,  in  exercise  8,  according  to  the  descending 
powers  of  x. 

Arrange  according  to  the  ascending  or  descending  powers  of 
some  letter  and  perform  the  multiplications  indicated : 

9.  ab(6a'  +  a*  +  l  +  4.a''-[-4:a). 

10.  2a;?/(8a^?/  +  2.T^  +  2?/*  +  12a^2/^  +  8a;/). 

11.  a26c(3  a'  + 16  6*  +  2  a¥  +  4  a«6  +  5  a-b'). 

12.  8  t^^  (t^  +  6  «s^  +  20  t^^  + 15  t\^  +  s«  + 15  <V). 

13.  5xy(a^-\-f  +  5x^y-{-5xy'^-i-10a^y--^10x'f). 


32  FUNDAMENTAL   OPERATIONS 

29.    To  multiply  a  polynomial  by  a  polynomial. 

EXERCISES 

1.  Multiply  x*  -j-  5  by  a;  4-  2 ;  test  the  result. 

PROCESS  TEST 

aj  4-  5  =6  when  x  =  l 

X  +2  =   3 

X  times  (x  -f-  5)  =  ic^  +  5  ic 
2  times  (x  +  5)  =         2  a; +  10 
(a;4-2)  times  (x-\-5)  =x--f  7a;+  10         =18 

Test.  —  The  product  must  equal  the  multiplicand  multiplied  by  the 
multiplier,  regardless  of  what  value  x  may  represent.  To  test  the  result, 
therefore,  we  may  assign  to  x  any  value  we  choose  and  observe  whether, 
for  that  value,  product  obtained  =  multiplicand  x  multiplier.  When 
X  =  1,  multiplicand  =  6,  multiplier  =  3,  and  a;'-  +  7  x  +  10  =  18  ;  since 
6  X  3  =  18,  it  may  be  assumed  that  a;^  +  7  x  +  10  is  the  correct  product. 

EuLE.  —  Multiply  the  multiplicand  by  each  term  of  the  multi- 
plier andjind  the  sum  of  the  partial  products. 

2.  Multiply  a;  +  4  by  a?  +  6  ;  test  the  result  when  x  =  l. 

3.  Multiply  a;  -f- 1  by  x-\-2;  test  the  result  when  x  =  6. 

4.  Multiply  2aj  +  3by4a;+l;  test  the  result  when  a;  =  1. 

5.  Multiply  a;^  +  aj  +  lbya;-f-l;  test  the  result  when  a;  =  2. 

6.  Multiply  2a  +  5  +  c  by  3  a-\-b;  test  the  result  -when 
a  =  l,  b  =  l,  and  c  =  1. 

In  like  manner  the  multiplication  of  any  two  literal  expressions  may  be 
tested  arithmetically  by  assigning  any  values  we  please  to  the  letters. 

While  it  is  usually  most  convenient  to  substitute  1  for  each  letter,  since 
this  may  be  done  readily  by  adding  the  numerical  coefficients,  the  student 
should  bear  in  mind  that  this  really  tests  the  coefficients  and  not  neces- 
sarily the  exponents,  for  any  power  of  1  is  1. 

Multiply,  and  test  each  result: 
7.    x-{-y  8.    a; +  4  9.    2a;4-l  10.    5y-\-3z 

x4-y  ar  +  9  3x-\-5  4?/+    z 


FUNDAMENTAL   OPERATIONS  33 

Multiply,  and  test  each  result : 

11.  2x-^3hj  x  +  2.  15.  3  I -^5  thy  21  +  6 1. 

12.  4  a; +  1  by  3  a; +  4.  16.  4:  y  +  6  b  by  2  y -\-b. 

13.  5?i  +  lby4w  +  5.  17.  2b  +  5  chy  5b-{-2  c. 

14.  h-\-2khy  3  h  +  k.  18.  ax  +  by  by  ax  +  by. 

An  indicated  product  is  said  to  be  expanded  when  the  multi- 
plication is  performed. 

Expand,  and  test  each  result : 

19.  (c'  +  (f)((^-\-(P).  23.  (^a-f  i&)(ia  +  i6). 

20.  (3a  +  b)(3a  +  b).  24.  (ix  + \y)(lx  + ^  y). 

21.  (2n'  +  I)(7v'  +  2l).  25.  (2  xy  +  3y)(4:xy -{- 7  y). 

22.  (2  6  +  5  c)  (3  6 -f  8  c).  26.  (4:  ax -\- 3  by) (4:  ax +3  by). 

Multiply,  and  test  each  result: 

27.    x-  +  2xy-\-y^  28.    a -{-b  +  c  29.    2^-h3.s  +  6 

x  +  y a^i^c  t-{-2s-\-l 

30.  a'  +  ay-\-y'hya  +  7j.  32.    (3  ^  +  n -fl)  (1 -h  ^  +  n). 

31.  a.-^  +  3  a;2 -I- a;  by  a; -f- 1.  33.    (7^+2  rs-ts^(r -{-8  + 1). 

34.  2a'-\-3b-  +  abhy3a-  +  4ab-\-5b^ 

35.  3  71^  +  3  m^  +  mn  by  m^  +  2  7nn^  +  m^^ii. 

36.  a^  +  o^  +  4  a^  4- a^H- a  by  a +  1. 

37.  31  ar^  +  27  ar  +  9  a;  +  3  by  3  a; +  1. 

38.  4a;3  4-3a;2y_,_5^^2_^(;^^by  5^,_l_gy^ 

39.  a^  +  b'^  +  c^  +  ab  +  ac  +  be  by  a  +  b  -\-  c. 

40.  m®  +  m^/i*  +  m^n*"'  +  m^Ai^  4-  n^^  by  m^  +  n^. 

41.  Multiply  rt«"  +  a^^b-'  +  a^"6^'=  +  ft*'  by  a^  +  62^ 

42.  Multiply  a;-"  -f  2a;'*.?/*'  + 1/^  by  a;^  +  2  a;"?/"  +  j/^. 

MILNE'S    IST    YR.    ALG. 3 


34  FUNDAMENTAL  OPERATIONS 


DIVISION 

30.  In  multiplication  two  numbers  are  given  and  their 
product  is  to  be  found.  In  division  the  product  of  two  num- 
bers and  one  of  the  numbers  are  given,  and  the  other  number 
is  to  be  found. 

Division  is  thus  the  inverse  of  multiplication. 

Thus,  3  X  4  =  12  illustrates  multiplication  ; 
but         12  -r-  4  =    3  illustrates  division,  the  inverse  process. 

31.  To  divide  a  monomial  by  a  monomial. 

Because      a^  ■  a^z^  a^+3  =  a^^ 

a^  -^  a^  =  a^-3  =  a\  (Subtract  exponents) 

Similarly,  because 

3a2.  5a3  =  (3  .  5)  a'+^^Wa^, 
15  a^  -  5  a^  =  (15  -  5)  a'-^  =  3  a\      (Divide  coefficients) 
The  quotient  may  be  obtained,  just  as  in  arithmetic,  by  re- 
moving equal  factors  from  dividend  and  divisor  by  cancellation, 
thus: 

15a^__3'^'^'^'^'a'a__o    2 
6a^  ~         ft'  fb'  d'  ^         ~ 

Again,     gl^^^^^-/x-;^>^.a.a-^-i^^^^.^ 

or  ?1^'  =  —  a«-363-2  ^  7  fj2^i  ^  7  ^25 

3  a'b'       3 

Hence,  for  division : 

32.  Law  of  exponents.  —  TJie  exponent  of  a  number  in  the  quo- 
tient is  equal  to  its  exponent  in  the  dividend  minus  its  exponent 
in  the  divisor. 

Since  a  number  divided  by  itself  equals  1,  a^  -^  a^*  =  a^~^  =  a^  =  !• 
that  is,  a  number  whose  exponent  is  0  is  equal  to  1 . 

33.  Law  of  coefficients.  —  TJie  coefficient  of  the  quotient  is 
equal  to  the  coefficient  of  the  dividend  divided  by  the  coefficient  of 
the  divisor. 


FUNDAMENTAL  OPERATIONS  35 


EXERCISES 

34.    Tell  quotients  quickly : 

1.   5)53                          2.   7  c^d^)35  c*d'  3.   2  d^)a\v 

5=^                                             Sc^cif  ^a'x 

4.   22)2f                          5.   3^-3^  6.   4«m«)4^mV 

12  g^ft-                     g     18a;V  ^     21  a6V 

*     4a62  *                      *     8af^^  '  '       7  b^     ' 

^Q     28  a^6-c                  ^^     16a^^  ^2     24^y?!. 

4  a6c                              4  a;2/'^2;  8  a;^2^ 

20a^6y                ^4     36<vV  ^^     3a6(a  +  &y 

■     4:a^bY  ' ,                  *      9aV    *  *       2(a  +  5) 


JUtW^  ^7      4  a^y^^  jg     2a\x-y)\ 

20  a^b(f'  '    S2xy-z'  '      a(x-yy 

35.  To  divide  a  polynomial  by  a  monomial. 

Dividing  as  in  arithmetic,  we  have 

1.  2)86  2)80  4-  fi  2)8  tens  +  6  units  2)8  ^  +  6  ^ 

43  40  +  3  4  tens +3  units     .         At-\-Su 

2.  Since,  §  27,  (a -^  b)  x  =  ax -\-  bx, 

if  aa;  +  bx  is  regarded  as  the  dividend  and  x  as  the  divisor, 

(ax  +  bx) -i- x  =:  a -{- b  ]  that  is, 

36.  The  quotient  of  a  polynomial  divided  by  a.  monomial  is 
equal  to  the  sum  of  the  partial  quotients  obtained  by  dividing  each 
term  of  the  polynomial  by  the  monomial. 

EXERCISES 

37.  1.  Divide  9  x^'f  + 15  x'lfz^  by  xy- ;  by  3  xy. 

PROCESS  PROCESS 

yy^9  x^y-  +  15  ayyV  3  xy)^  xry"^  +15  xy^z^ 

9  if     +15  2-  SiC2/   +   5  2/2- 


36  FUNDAMENTAL   0PERAT10x\8 

Find  quotients : 
2.   4.  cd)  4:  c'dj- 20  cd^  7.   7  ab)U  a'b^ -{- Ad  a% 

g    xz'  +  3xz-{-x^z\  g    35  x'fz'  +  45  xYz^ 

xz  5  xryh 

^    5a^y  +  10a^/  +  15a;/         ^    36 a^6V  +  60 a^6V 
5xy  '  '  12  a'h'c^ 

4.a'W-{-12a%^^lQ,a^h      ,^    24  ?V -f  30  rV  +  42  ^-^^^ 

5.    ^ •      lu.    • 

4  a^6  6  ?-^s^ 

g    24a«6^+32a^6g+40a^6^     ^^    9  a.-^yg  +  36  a;y  V  +  45  a;yg^ 
8a*62  *        •  ^xyz 

12.  (8a'63  +  28a«6^  +  16a'^>'  +  4a*6^)^4a^63. 

13.  (3o?yz''-{-l^xY^-\-^x^y^  +  l%xYz)^3;x?yz. 

38.   To  divide  a  polynomial  by  a  polynomial. 


EXERCISES 

1.   Divide  3  ar^  +  35  +  22  .t  by  a;  +  5 ;  test  the  result. 


PROCESS 

TEST 

3a;2  +  22aj  +  35 

a;  4-5 

60-6 

3  X  times  (a;  +  5) 

33.-2  +  15  0; 

7a.'  +  35 

3a;  +  7 

=  10 

7  times  (x  +  5) 

7a;  +  35 

Explanation.  — For  convenience,  the  divisor  is  written  at  the  right  of 
the  dividend  and  the  quotient  below  the  divisor.  Both  dividend  and 
divisor  are  arranged  according  to  tlie  descending  powers  of  x. 

Since  the  dividend  is  the  product  of  the  quotient  and  divisor,  it  is  the 
sum  of  all  the  partial  products  formed  by  multiplying  each  term  of  the 
quotient  by  each  term  of  the  divisor.  Hence,  if  3  x^,  the  first  term  of 
the  dividend  as  arranged,  is  divided  by  x,  the  first  term  of  the  divisor, 
the  result,  3  a;,  is  the  first  term  of  the  quotient. 

Subtracting  3  x  times  {x  +  5)  from  the  dividend,  leaves  7  x  +  35,  the 
part  of  the  dividend  still  to  be  divided. 


V 


FUNDAMENTAL  OPERATIONS  37 

Proceeding,  then,  as  before  we  find,  1x-r-x  =  l,  the  next  term  of  the 
quotient.  7  times  {x  +  5)  equals  7  x  +  35.  Subtracting,  we  have  no 
remainder.     Hence,  the  quotient  is  3  x  +  7. 

Test.  —  When  x  =  1 ,  the  dividend  equals  60  and  the  divisor  6.  The 
quotient  then  should  equal  60  -h-  6,  or  10.  On  substituting  1  for  x,  we 
find  that  the  quotient  is  equal  to  10.  Presumably,  then,  the  result  is 
correct. 

2.    Divide  3.-3+ 6  a;- +  12  a; +  10  by  a; +  2. 


PRO 

a;3  +  6ar^  +  12a;  +  10 

CESS 

a;  +  2 

TEST 
29-r-3 

4a;^  +  12a; 

^+4. +4+^^^ 

=H 

4x^4.   8a; 

4a;  +  10 

4a;+   8 

\ 

As  in  arithmetic,  the  whole  of  the  undivided  part  of  the  dividend  is  not 
brought  down  for  each  division,  but  only  so  much  of  it  as  may  be  needed 
each  time. 

The  remainder  2  is  written  over  the  divisor  in  the  form  of  a  fraction 
which  is  then  added  to  the  quotient  as  in  arithmetic. 

Rule.  —  Airange  both  dividend  and  divisor  according  to  the 
ascending  or  the  descending  powers  of  a  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  ivrite  the  result  for  the  first  term  of  the  quotient. 

Multijyly  the  whole  divisor  by  this  term  of  the  quotient,  and  sub- 
tract the  product  from  the  dividend.  Tlie  remainder  ivill  be  a 
new  dividend. 

Divide  the  new  dividend  as  before,  and  continue  to  divide  in 
this  way  until  the  first  term  of  the  divisor  is  not  contained  in  the 
first  term  of  the  new  dividend. 

If  there  is  a  remainder  after  the  last  division,  write  it  over  the 
divisor  in  the  form  of  a  fraction,  and  add  the  fraction  to  the  part 
of  the  quotient  previously  obtained. 


38  FUNDAMENTAL   OPERATIONS 

Divide,  and  test  each  result : 

3.  x--{-2x  +  lhj  x-\-l.  6.   3 -h7/  +  2?/*  by  2/2-1-3. 

4.  a'-h5a-\-6hj  a-\-2.  7.    6c(^ -\-af -{-7  hy  x^ -]-l. 

5.  5r  +  r2  +  4by  r  +  4.  8.    G^^.,.  20^  +  23  by  3^ -|- 7. 
9.  .v^-h3/  +  32/  +  lby2/  +  l. 

10.  (jz'-h4.  +  10z^-\-4.:^hj  4:z'-\-2. 

11.  b^  +  Cyb^-\-b'-{-9b^-\-4.b-\-Shjb^-^4:. 

12.  Dividea*-h6a'^  +  27a2-h54a-f81  by  a^ -h  3  a -1- 9. 


PROCESS 

TEST 

a4^(5^3^27a2-f-o4a-h81 

a^  -h  3  a  +  9 

169-13 

a*  +  3a^+    9a^ 

a^  +  3a  +  9 

=  13 

3a'  +  lSa'  +  54.a 

Sa^-\-   9a2  +  27a 

9a-  +  27a-|-81 

9a2  4-27a-f-81 

Divide,  and  test  each  result  : 

13.  x'-\-A^i-12:t^-j-16x+16hy  x^-\-2x-{-4:. 

14.  4:l'-\-4.l'  +  13l*  +  6l'-j-9hj2l'  +  P  +  3. 

15.  42/^-|-52/'  +  /4-ll2/H-3/  +  6  by  .v^  +  3.v  +  2. 

16.  6r^ -f-26r2  +  18-hl5r+7r3by2r-f-3r2-h3. 

17.  X'  -h  .<v  +  2  xY  -{-2xY-\-  xy^  4-  2/^  by  a?  -}-  2/. 

18.  2  a^  -h  6  o^  4-  3  a2  -f-  2  a^  4-  ^"5  a  -f-  2  by  a^  +  a  -h  2. 

19.  aj^  4-  2  i»«?/  +  4  ar"^?/-  4-  3  xhf  -\- 2  3?y^  hj  x^ -\- xy  +  f, 

20.  / 4- 8 z« 4-3 2^4- 2^^  +  6:^4-20^^  +  30  by  2! 4-32^ +  5. 

21.  hsH■^^s^f'^t'-\-3s'^^38fhy3s^^2sH^2st^^-f. 

22.  4a^  +  28a-^6  +  61a262  +  45a63  +  12  6^by2a2  +  7a6  +  3  62. 

23.  j5«  +  4pV  +  9pY  +  6  9«  +  2j7'g  +  6pY  +  12pg'  by  p^  + 
3pV  +  65^. 


FUNDAMENTAL   OPERATIONS  39 


EQUATIONS   AND    PROBLEMS 

39.  How  many  pounds  added  to  25  pounds  will  give  30 
pounds  ? 

The  statement  of  the  problem  may  be  condensed  to 

25  pounds  25  pounds 

+  ?  pounds       or  +_x^  pounds     or   25  +  a;  =  30 
30  pounds  30  pounds 

.  The  letter  x  is  only  a  convenient  symbol  for  the  unknown 
number  (of  pounds),  or  the  number  (of  pounds)  to  be  found. 
25  and  30,  on  the  other  hand,  are  known  numbers. 

The  equation,  25  +x  =  30,  is  the  briefest  possible  statement 
of  the  relation  between  the  known  and  unknown  numbers  in 
the  problem.  Finding  the  value  of  x  is  called  solving  the 
equation,  25  -f-  a;  is  the  first  member  of  the  equation,  and  30  is 
the  second  member. 

Equations 

40.  1.  If  25  pounds  are 
taken  from  the  weight  in 
each  scale  pan,  the  bal- 
ance will  be  preserved. 

In  the  same  way,  if  25 
is  subtracted  from  each  member  of  the  equation  25  +  a;  =  30, 
the  equality  will  be  preserved. 

25  +  a;  =  30 

25 25 

x=  5 

2.   What  number  subtracted  from  x  -f  10  will  give  x  ? 

If  the  first  member  of  a;  + 10  =  12  is  decreased  to  x  by  sub- 
tracting 10,  what  must  be  done  to  the  second  member  to  pre- 
serve the  equality  ? 

Tell  how  the  equation  a;  -|-  10  =  12  may  be  solved. 


40  FUNDAMENTAL  OPERATIONS 

3.  Suppose  that  a;  —  4  =  3  and  we  wish,  to  find  the  value  of 
X.     How  much  greater  is  x  than  a?  —  4  ? 

If  the  first  member  of  a;  —  4  =  3  is  increased  to  x  by  adding 
4,  what  must  be  done  to  the  second  membeiv  to  preserve  the 
equality  ?     Tell  how  the  equation  may  be  solved. 

The  same  number  may  he  added  to  both  members  of  an  equa- 
tioHy  or  subtracted  from  both,  without  destroying  the  equality. 

EXERCISES 

41.  State  what  must  be  done  to  both  members  to  change  one 
member  to  x  without  destroying  the  equality  ;  solve  : 

1.  x  +  6==S.  5.  0^4-2  =  10.  9.  12  =  10  +  a;. 

2.  a; -3  =  2.  6.  a;- 5  =  11.  10.  15  =  11  + a;. 

3.  a;-4  =  5.  7.  a;  +  l  =  12.  11.  30  =  20  +  aj. 

4.  a; 4- 7  =  9.  8.  a;— 7  =  10.  12.  14  =  a; +  10. 

42.  1.    If  a;  =  8,  what  is  the  value  of  2a;?  of  3a;?  of  fa;? 

2.  If  6  a;  =  12,  what  is  the  value  of  1  a;,  or  of  a;  ? 

3.  If  ^a;  =  10,  what  is  the  value  of  3  times  ^x,  or  of  a;  ? 

4.  What  must  be  done  to  both  members  of  each  of  the  fol- 
lowing equations  to  give  an  equation  whose  first  member  is  a;? 

|a;  =  3         lx  =  5         4a;  =  12         5a;  =  35 

Both  members  of  an  equation  may  be  multiplied  or  divided  by 
the  same  number  without  destroying  the  equality. 

EXERCISES 

43.  State  what  must  be  done  to  both  members  to  change 
one  member  to  x  without  destroying  the  equality ;  solve : 

1.  2x=Q.  5.  |a;  =  5.  9.  \x  =  6. 

2.  5  a;  =  5.,  6.  ^-a;  =  2.  10.  ia;  =  4. 

3.  4^^.  7.  ia;  =  3.  11.  8  a;  =  24. 
4^/3a;  =  9.  8.  ^a;  =  7.  12.  9a;  =  18. 


FUNDAMENTAL   OPERATIONS  41 

44.  The  equations  solved  so  far  in  this  chapter  have  been 
solved  each  by*  a  single  one  of  the  following  steps : 

1.  By  adding  the  same  number  to  both  members. 

2.  By  subtracting  the  same  number  from  both  members. 

3.  By  multiplying  both  members  by  the  same  number. 

4.  By  dividing  both  members  by  the  same  number. 

The  equations  that  follow  may  be  solved  by  two  or  more  of 
these  steps  taken  separately. 

EXERCISES 

45.  1.   Solve  the  equation  2  a;  +  20  =  80  -  4  x. 

SOLDTION 

The  first  step  in  solving  an  equation  is  to  get  the  unknown  terms  into 
one  member,  usually  the  first,  and  the  known  terms  into  the  other. 
2  X  +  20  =  80  -  4  x. 
Adding  4  x  to  both  members, 

2x  +  4x  +  20  =  80  +  4x-4x, 
or,  uniting  terms,  6  x  +  20  =  80. 

SuWacting  20  from  both  members, 

6  x-f  20- 20  =  80 -20, 
or,  uniting  terms,  6  x  =  60. 

Dividing  both  members  by  6,    x  =  10. 

Verification 

We  should  always  test  the  answer  by  finding  whether  the  value 
obtained  is  such  as  to  make  the  members  of  the  original  equation  equal. 

Thus,  substituting  x  =  10  in  the  given  equation,  we  have 
20  +  20  =  80  -  40, 
or  40  =  40. 

Hence,  10  is  the  true  value  of  x. 

Solve  and  verify : 

2.    7x-\-12  =  5x-[-W.  6.  4a;-ll  +  2a;  =  2a;-5. 

^3.   5a;-20  =  2a;  +  13.  7.  Sx-\-U-^7 x  =  7S-\-2x. 

4.  4a;-ll  =  19-2a;.  8.  9  a;  + 23  +  2a;  =  4  .^  +  37. 

5.  13a;  +  4  =  5a;  +  12.  9.  5  a; -7 +  15  a;  =13  a; +14. 


42  FUNDAMENTAL  OPERATIONS 

10.  Solve  the  equation  ^  x=  15. 

First  Solution 

fa;  =  15. 
Dividing  both  members  by  3,  ^x=  6. 
Multiplying  both  members  by  2,      x  =  10. 

Second  Solution 

By  multiplying  by  2  before  dividing  by  3,  fractions  may  be  avoided. 

fx  =  16. 
Multiplying  both  members  by  2,  Sx  =  30. 
Dividing  both  members  by  3,  x  =  10. 

Verification.    |  of  10  =  16. 

Solve  and  verify : 

11.  fa;  =  9.  13.   fa;  =  21.  15.   |a;  =  15. 

12.  |a;  =  8.  14.    fx  =  30.  16.   |a;=21. 

Solve  by  the  method  best  adapted ;  verify  results  : 

17.  9x-lT  =  23-{-x.  23.   fa;  =10. 

18.  2a;  +  3a;-2x  =  21.  24.   |a;  =  14. 

19.  9x-4:X  +  2x  =  U.  25.   |£c  =  28. 
20:  Sx-\-5x  —  5x  =  AS.  26.   |«  =  20. 

21.  22-6x  =  ^0-Sx.  27.   |a;  =  63. 

22.  7  a;  +  6  a;  -  7  a;  =  42.  28.   f  a;  =  48. 

29.  2a;-4  +  6a;  =  22-15  +  21. 

30.  5a;-10-4x=46  +  3x-60. 

31.  6aj+5.T-70  =  5a.'  +  54-70. 

32.  5a;  +  16-6a;  =  16  +  24-6a.'. 

33.  9aj  +  15-2a;  =  32+4a;-ll. 

34.  10x-39  +  12a;-9a;-t-42-4a;  =  42-4ar. 

35.  1 6  a;  + 12  -  75  +  2  a;  -  12  - 110  =  8  aj  -  50  -  25. 

36.  3  a;  - 18  +  27  +  10  a;  -  11  =  25  4-  4  a;  -  7  a;  + 12  -h  3  a;. 

37.  18  .a;  + 16  =  8  + 12  aj  +  8  - 13  -f  25  a;  -  9  +  100  -  25  a;. 


OF  THE 

UNIVERSITY 

Tndamental  operations  43 


Algebraic  Representation 
46.    1.    Express  the  sum  of  2,  i,  and  ^ ;  of  x,  ^  y,  and  J  z. 

2.  What  number  is  4  less  than  12?  n  less  than  25  ? 

3.  Express  the  number  that  exceeds  5  by  3 ;  ahj  b. 

4.  Represent  in  the  shortest  way  the  sum  of  five  ic's ;  the 
product  of  five  x's. 

5.  Mary  read  10  pages  of  a  book.  On  what  page  did  she 
begin  to  read,  if  she  stopped  at  the  top  of  page  21  ?  of  page  a  ? 

6.  Express  10  dollars  in  terms  of  cents ;  10  cents  in  terms 
of  dollars ;  m  dollars  in  terms  of  cents ;  m  cents  in  terms  of 
dollars. 

7.  A  has  12  dollars  and  B,  8  dollars.  How  much  will  each 
have  if  A  gives  B  4  dollars  ?     m  dollars  ? 

8.  At  3  dollars  per  day,  how  much  will  a  man  earn  in  4 
days?  in  x  days?  At  a  dollars  per  day,  how  much  will  he 
earn  in  b  days  ?  in  c  days?  in  a  days  ? 

9.  By  what  number  must  25  be  multiplied  to  produce  300  ? 
10  to  produce  x?  r  to  produce  s  ? 

10.  What  are  the  two  odd  numbers  nearest  to  5  ?  If  ?i  +  3 
is  an  odd  number,  what  are  the  two  odd  numbers  nearest  to 
ri  +  3  ?  the  two  even  numbers? 

11.  How  many  square  rods  are  there  in  a  square  field  one 
of  whose  sides  is  2  rods  long  ?  (x-\-y)  rods  long  ? 

12.  How  many  square  rods  are  there  in  a  field  6  rods  long 
and  4  rods  wide  ?     (m  -h  n)  rods  long  and  m  rods  wide  ? 

13.  If  it  takes  4  men  5  days  to  do  a  piece  of  work,  how  long 
will  it  take  1  man  to  do  it?  2  men?  x  men?  If  it  takes  b 
men  c  days  to  do  a  piece  of  work,  how  long  will  it  take  1  man  ? 
z  men  ? 

14.  The  number  25  may  be  written  20  +  5.  Write  the 
number  whose  first  digit  is  x  and  second  digit  y. 

15.  Represent  (a  +  b)  times  the  number  whose  tens'  digit  is 
m  and  units'  digit  n. 


44  FUNDAMENTAL   OPERATIONS 

Problems 
47.    1.    What  number  increased  by  6  is  equal  to  44  ? 

Solution 

Let  %  =  the  number. 

Then,  '  a; +  6  =  44. 

Solving  the  equation,  x  =  38,  the  number. 

2.  What  number  increased  by  15  is  equal  to  51  ? 

3.  What  number  decreased  by  32  is  equal  to  60  ? 

4.  What  number  multiplied  by  3  is  equal  to  78  ? 

5.  What  number  divided  by  8  is  equal  to  62  ? 

6.  If  20  is  added  to  a  certain  number  and  14  is  subtracted 
from  the  sum,  the  result  is  19.     Find  the  number. 

7.  One   half  of   a  number,  and  11   more,  is  equal  to  37. 
Find  ^  of  the  number,  then  find  the  number. 

8.  If  f  of  a  certain  number  is  18,  what  is  the  number  ? 

9.  The  sum  of  two  numbers  is  55  and  the  larger  is  4  times 
the  smaller.    What  are  the  numbers? 

Solution 

Let  X  =  the  smaller  number. 

Then,  4  a;  =  the  larger  number, 

and  cc  4-  4  X  =  the  sum  of  the  two  numbers. 

But  55  =  the  sum  of  the  two  numbers. 

.'.  aj  +  4ic  =  55. 

Solving  the  equation,        a;  =  11,  the  smaller  number, 
and  •     4  X  =  44,  the  larger  number. 

Note.  —  The  sign  .-.  means  '  therefore.' 

10.  Separate  116  into  two  parts,  one  of  which  shall   be  3 
times  the  other. 

11.  Separate  72  into  two  parts,  one  of  which  shall  be  ^  of 
the  other. 

12.  What  number  increased  by  ^  of  itself  equals  54  ? 

13.  What  number  decreased  by  |  of  itself  equals  84  ? 


FUNDAMENTAL   OPERATIONS  45 

14.  Five  times  a  number  exceeds  3  times  the  number  by  14. 
What  is  the  number  ? 

15.  The  double  of  a  number  is  64  less  than  10  times  the 
number.     What  is  the  number? 

16.  Four  times  a  certain  number  exceeds  12  as  much  as  3 
times  the  number  is  less  than  72.     What  is  the  number  ? 

17.  Of  the  steam  vessels  built  on  the  Great  Lakes  one  year, 
21,  or  5  less  than  ^  of  all,  were  of  steel.  How  many  steam 
vessels  were  built  on  the  Lakes  that  year  ? 

Solution 
Let  X  =  the  number  of  steam  vessels  built. 

Then,  J  x  —  5  =  the  number  of  steel  vessels. 

But  21  =  the  number  of  steel  vessels. 

...  l-x-5  =  21. 
Adding  5  to  both  members  of  the  equation, 
|a;  +  6-5  =  21  +  6, 
or  ^x  =  26. 

Multiplying  both  members  by  3, 

X  =  78,  the  number  of  steam  vessels  built. 
Note.  —  The  equation  ^  a;  —  5  =  21  is  called  the  equation  of  the  problem. 

General  Directions  for  Solving  Problems.  —  1.  Represent  one 
of  the  unknown  numbers  by  some  letter,  as  x. 

2.  From  the  conditions  of  the  problem  find  an  expression  for 
each  of  the  other  unknown  numbers. 

3.  Find  from  the  conditions  two  expressions  that  are  equal 
and  write  the  equation  of  the  problem. 

4.  Solve  the  equation. 

18.  Two  cars  together  contained  400  bales  of  cotton.  If 
one  car  had  6  bales  more  than  the  other,  how  many  had  each  ? 

19.  The  playgrounds  of  two  cities  occupy  183  acres.  One 
city  has  27  acres  less  than  the  other.  How  many  acres  has 
each  ? 

20.  The  height  of  the  big  tree  Wawona  in  California  is 
8  feet  more  than  9  times  its  diameter.  If  the  height  is  260 
feet,  what  is  the  diameter  of  the  tree  ? 


46  FUNDAMENTAL   OPERATIONS 

21.  Yellowstone  Park  contains  3400  antelope  and  deer.  If 
the  antelope  number  200  less  than  twice  the  number  of  deer, 
how  many  deer  are  there  in  the  Park? 

22.  A  department  store  restaurant  serves  luncheon  daily  to 
6000  people.  If  the  number  served  lacks  1000  of  being  3  times 
the  number  seated  at  once,  find  the  seating  capacity. 

23.  One  year  the  Bureau  of  Engraving  and  Printing  em- 
ployed 2400  people.  The  number  of  women  was  400  greater 
than  the  number  of  men.     Find  the  number  of  each  employed. 

24.  In  a  recent  year  the  Lake  Superior  region  furnished 
38,400,001)  tons  of  iron  ore,  or  |  of  all  that  was  mined  in  the 
United  States.     How  much  was  mined  in  the  United  States  ? 

25.  Denmark  produces  44,000  tons  of  beet  sugar  annually. 
If  this  is  4000  more  than  i  the  number  of  tons  consumed,  what 
is  the  annual  consumption  of  beet  sugar  in  Denmark  ? 

26.  One  year  the  government  spent  $  60,000  in  operating  a 
flag  factory.  The  material  cost  $  8000  less  than  3  times  the 
amount  expended  for  labor.     What  was  the  cost  of  each  ? 

27.  Two  power  companies  together  use  27,200  cubic  feet  of 
water  per  second  from  Niagara  Falls.  Find  the  average  dis- 
charge of  the  falls  per  second,  if  these  companies  use  -^  of  it. 

28.  The  whalebone  in  one  whale  was  worth  y^  as  much  as 
that  in  another,  and  the  value  of  the  whalebone  in  the  two 
was  $  525.     Find  the  value  of  the  whalebone  in  each. 

29.  In  a  fossil  bed  in  Switzerland  470  species  of  insects 
were  found  and  this  was  30  less  than  |  of  the  number  found 
in  a  bed  in  Colorado.     Find  the  number  in  the  latter  bed. 

30.  The  largest  cask  in  the  world  contains  a  number  of  hogs- 
heads that  is  1  less  than  25  times  the  number  of  feet  in  its  di- 
ameter.    If  it  contains  649  hogsheads,  find  its  diameter. 

31.  The  railroads  consume  ^^  of  the  total  annual  production 
of  coal  in  the  United  States.  Their  annual  expenses  for  coal 
are  240  million  dollars  with  the  average  price  ^2  per  ton. 
How  many  tons  are  produced  in  the  United  States  each  year  ? 


FUNDAMENTAL  OPERATIONS  47 

32.  The  United  States  uses  101  million  files  each  year.  The 
number  of  files  made  in  this  country  lacks  15  million  of  being 
3  times  the  number  of  those  imported.  How  many  files  are 
imported  ? 

33.  The  Jamestown  Exposition  pier  inclosed  a  rectangular 
lagoon,  the  length  of  which  was  1000  feet  more  than  its  width. 
If  its  perimeter  was  6800  feet,  how  long  was  it  ? 

34.  In  one  year  the  output  of  scrap  mica  was  5  tpns  more 
than  twice  the  output  of  sheet  mica  and  there  were  430^  tons 
more  of  the  former  than  of  the  latter.  Find  the  number  of 
tons  of  each. 

35.  The  largest  concrete  chimney  in  the  world  contains 
1460  tons  of  steel  and  sand.  The  weight  of  the  steel  used  was 
7^  of  the  weight  of  the  sand.  Find  the  number  of  tons  of 
each  that  were  used. 

36.  Mt.  Whitney,  the  highest  point  in  the  United  States, 
is  14,500  feet  above  sea  level.  This  is  700  feet  more  than  50 
times  the  depth  below  sea  level  of  Death  Valley,  the  lowest 
point  of  dry  land  in  the  country.  How  far  below  sea  level  is 
Death  Valley  ? 

37.  The  lilies  sent  to  the  United  States  annually  from  Ber- 
muda are  worth  -^  as  much  as  all  our  imported  floral  products. 
If  the  other  floral  products  are  worth  S  1,900,000,  find  the 
value  of  the  lilies  imported  from  Bermuda. 

38.  The  distance  from  Cuba  to  Haiti  is  31  miles  less  than 
the  distance  to  Jamaica,  and  from  Cuba  to  Yucatan,  which  is 
130  miles,  is  9  miles  less  than  the  sum  of  the  distances  to  Haiti 
and  Jamaica.     Find  the  distance  from  Cuba  to  Jamaica. 

39.  In  field  and  track  events  at  the  Olympic  games  in  Lon- 
don, America  won  35|-  points  more  than  Great  Britain  and 
Sweden  together,  and  Sweden  won  54  points  less  than  Great 
Britain.  Find  the  score  of  each,  if  the  total  score  of  the  three 
countries  was  1931  points. 


48  REVIEW 

REVIEW 

48.    1.    Tell  how  similar  terms  are  added  ;  subtracted.     Tell 
what  to  do  with  dissimilar  terms  in  addition ;  in  subtraction. 

2.  Write  a  polynomial  arranged  according  to  the  ascending 
powers  of  some  letter ;  the  descending  powers. 

3.  State  the  law  of  exponents  for  multiplication ;  for  divi- 
sion ;  the  law  of  coefficients  for  each.     3^  =  ?     8^  =  ?     a"  =  ? 

4.  What  is  an  equation  ?  Write  one  and  point  out  the 
unknown  numbers  in  it ;  the  known  numbers ;  its  first  member ; 
its  second  member. 

5.  What  is  meant  by  ^  solving  an  equation '  ?  Give  four 
methods  by  one  or  more  of  which  equations  may  be  solved. 
How  may  the  value  of  an  unknown  number,  obtained  by  solv- 
ing an  equation,  be  verified  ? 

Solve  and  verify  : 

6.  3a;  =  21.  8.   7  x-3-\-4:X  =  21-2  x-^2. 

7.  I  a;  =  15.  9.   10-2a;  =  3x  +  5  +  9-6a;. 

10.  Add  X  -{-  y  +  z,  7  x+  2  z-h3  y,  4  z-^5  y,  and  9  a;+3  y-{-2  z. 

11.  11a -f5  6  +  2  c- 4  6-f2  a -c+4c-9a4-5  6-3  c+a=? 

12.  Subtract  5  a**  H-  7  6"  +  18  c. from  7  a"*  -|-  25  c  -f  8  6"  +  8  d 
Expand : 

13.  7p^^r^(pqr^+4:p-qr  +  2qi^+p^  +  5pY-r*-\-Spq). 

14.  (x*  -f-  7  a;'?/  +  4  a^2/'  +  3  xy^  +  2  y*)(x^  +  ^'^-y  +  2  f). 

15.  (3  z^  +  4  z^w  +  6  z^w"  +  4  2w;3  +  13  w%2  z'  +  4ziv-\-S  w"). 

16.  {a'h  +  3  a^'W  +  6  a^W  +  5  a^W  + 11  ah'){a?  -j-  3  a^^  -f  4  aW), 
Divide,  and  test  each  result : 

17.  12  ZVn2  -f  18  fm'n^  -f  15  ZVn^  +  3  J^mn"  by  3  l^mn\ 

18.  35  r*  -f  30  r's  +  69  rh^  +  12  r^  +  22  s^  by  5  r^  +  2  s\ 

19.  22  xY  +  24  xy^  -f  27  o^f  +  36  a;y  +  3  a;^?/  by   3  a^?/  +  4  yK 

20.  4  a^c  +  8  a^hc  + 11  a^Wc  +  24  a^b^c  4-  24  ah^c  -f-7  h'c  by  2  «2 
-f  3  a6  +  61 


POSITIVE   AND   NEGATIVE   NUMBERS 


49.  The  student  of  arithmetic  knows  the  meaning  of  such 
an  expression  as  10  —  4,  but  as  yet  an  expression  like  4—10 
has  no  meaning  to  him.  It  is  the  purpose  of  this  chapter  to 
extend  the  idea  of  number  so  that  subtracting  a  larger  number 
from  a  smaller  one  will  have  as  much  meaning  as  subtracting 
a  smaller  number  from  a  larger  one,  to  show  that  there  is  a 
practical  demand  for  a  new  kind  of  number,  and  finally  to 
show  how  operations  involving  this  new  kind  of  number  are 
performed. 

50.  Suppose  that  at  noon  the  temperature  is  10°  above  0 
and  that  at  6  p.m.  it  has  fallen  4°.  The  temperature  is  then 
10°  -4°,  or  6°  above  0,  but  if  it  has  fallen  15°  instead  of  4°,  it 
is  then  10°  — 15°,  and  because  the  numbers  on  a  thermometer 
extend  below  as  well  as  above  0,  we  see  that  10°  — 15°  means 
that  the  temperature  is  5°  below  0,  10°  of  the  15°  of  fall  taking 
it  to  0  and  the  other  5°  of  fall  taking  it  to  5°  below  0. 

For  convenience  and  brevity  degrees  '  above  0 '  are  marked 
with  the  sign  -|-  and  degrees  '  below  0 '  with  the  sign  — . 
Such  statements  may  be  abbreviated  algebraically,  thus : 

+  10° -4°  =  +  6°, 
and  + 10°  - 15°  =  -  5°. 

Similarly,  if  a  ship  now  at  20°  north  latitude  (latitude,  +  20°) 
sails  south  30°,  it  will  cross  the  equator  (latitude,  0°)  and  be  at 
10°  south  latitude  (latitude,  -  10°). 

Again,  a  tourist  in  going  from  Lake  Lucerne  1435  feet  above 
sea  level   (altitude    +1435   feet)  to  the  Dead  Sea  1295  feet 
below  sea  level  (altitude  —  1295  feet)  goes  not  only  to  0  alti- 
tude (sea  level),  but  ^/irojfj^/i  0  altitude. 
milne's  1st  yr.  alg. — 4  49 


60  POSITIVE   AND   NEGATIVE  NUMBERS 

51.  Such  illustrations  as  those  on  the  preceding  page  show 
a  practical  need  of  extending  the  number  scale  of  arithmetic, 

1,  2,  3,  4,  5,  ...,* 

below  0,  as  on  the  thermometer. 

The  scale  of  algebraic  numbers,  then,  including  0,  is 

...,  -5,  -4,  -3,  -2,  -1,0,  +1-,  +2,  +3,  +4,  +5,..., 

the  numbers  in  the  scale  ini3reasing  by  1  from  left  to  right. 

52.  Numbers  greater  than  0,  called  positive  numbers,  are 
written  either  with  or  without  the  sign  +  prefixed. 

Numbers  less  than  0,  called  negative  numbers,  always  have 
the  sign  —  prefixed. 

53.  By  repeating  the  positive  unit,  + 1,  any  positive  integer 
may  be  obtained,  and  by  repeating  the  negative  unit,  —  1,  any 
negative  integer  may  be  obtained. 

Fractions  are  measured  by  positive  or  negative  fractional  units. 

54.  It  is  seen,  then,  that  while  in  arithmetic  the  signs  4- 
and  —  are  used  to  indicate  operations  to  be  performed,  they 
have  an  extended  meaning  and  use  in  algebra,  namely,  to  de- 
note opposition.  In  this  sense  they  are  called  quality,  or 
direction,  signs. 

Thus,  if  gains  are  considered  positive,  indicated  by  +,  losses  are  nega- 
tive, indicated  by  —  ;  if  credits  are  +,  debts  are  —  ;  if  distances  north  or 
west  ov  upstream  are  +,  distances  south  or  east  or  downstream  are  — . 

55.  When  it  is  necessary  to  distinguish  between  signs  of 
operation  and  signs  of  quality  the  number  with  its  sign  of 
quality  may  be  inclosed  in  parentheses. 

Thus,  the  sura  of  2  and  —  3  may  be  written  2  +(—  3). 

56.  Positive  and  negative  numbers,  whether  integers  or 
fractions,  are  called  algebraic  numbers. 

Arithmetical  numbers  are  positive  numbers. 

*  The  sign  of  continuation,  •••,  is  read  '  and  so  on  '  or  '  and  so  on  to.' 


POSITIVE   AND  NEGATIVE   NUMBERS  51 

57.  The  value  of  a  number  without  regard  to  its  sign  is 
called  its  absolute  value. 

Thus,  the  absolute  value  of  both  +  4  and  —  4  is  4. 

ADDITION   AND    SUBTRACTION 

58.  Addition  and  subtraction  of  positive  and  negative  num- 
bers may  be  performed  by  counting  along  the  scale  of  algebraic 
numbers. 

To  illustrate,  +  3  is  added  to  —  2  by  beginning  at  —  2  in  the  scale  and 
counting  3  units  in  the  ascending^  or  positive,  direction,  arriving  at  +  1 ; 
consequently,  +  3  is  subtracted  from  +  1  by  beginning  at  +  1  and  count- 
ing 3  units  in  the  descending,  or  negative,  direction,  arriving  at  —  2. 

59.  The  result  of  adding  two  or  more  algebraic  numbers  is 
called  their  algebraic  sum. 

This  differs  from  their  arithmetical  sum,  which  is  the  sum  of  their 
absolute  values. 

Unless  otherwise  specified  '  sum  '  in  this  book  means  *■  algebraic  sum '. 

60.  In  addition,  two  numbers  are  given,  and  their  algebraic 
sum  is  to  be  found.  In  subtraction,  the  algebraic  sum  and  one 
of  the  numbers  are  given,  and  the  other  number  is  to  be  found. 

Subtraction  is  thus  the  inverse  of  addition. 
The  difference  is  the  algebraic  number  that  added  to  the  sub- 
trahend gives  the  minuend. 

Sum  of  Two  or  More  Numbers 

EXERCISES 

61.  Give  algebraic  sums  : 

1.    +5  -5  -f-5  +5  +5         -5         -5 

4-5  -5  -5  -4  -9  -f8  +2 

Suggestions.  —  The  sum  of  5  positive  units  and  5  positive  units  is  10 
positive  units ;  of  5  negative  units  and  5  negative  units,  10  negative  units  ; 
of  5  negative  units  and  5  positive  units,  0  ;  of  4  negative  units  and  6  posi- 
tive units,  1  positive  unit ;  of  -  9  and  +5,  —  4  ;  etc. 


52 


POSITIVE   AND   NEGATIVE  NUMBERS 


To  add  two  algebraic  numbers : 

Rule.  —  If  they  have  like  signs,  add  the  absolute  values  and 
prefix  the  common  sign;  if  they  have  unlike  signs,  find  the  differ- 
ence of  the  absolute  values  and  iwefix  the  sign  of  the  numerically 
greater. 

By  successive  applications  of  the  above  rule  any  number  of  numbers 
may  be  added. 


6.-5 

±^ 

11.    -9 
3 

2 


2.    8 

2 

3.    +8 
+  2 

4.    ^8 
-2 

5.    -h4 

-7 

7.        6 
-3 

7 

8.    -h7 
-3 

+  2 

9.-5 
-3 
-8 

10.    H-8 
-9 

+  1 

45 


-8 


12.  10-[-(-4)-h(-6)-h(-7).     15.    _12-h8-f-2  4-(-6)-f2. 

13.  10-4-6-7.  16.    -12-h8-h2-6H-2. 

14.  _40  +  6-h8  +  7-h6.  17.   8-2H-3-h6-8-h7-9. 

18.  Julius  Caesar  was  born  in  the  year  —  100  (100  b.c),  and 
was  5Q  years  old  when  assassinated.  In  what  year  was  he 
assassinated  ? 

19.  In  a  football  game  the      ^  +5  ^ 
ball    was    advanced    5    yards       !  "- 

from  the  Juniors'  25-yard  line     2.1       l +i3 

toward  the  Seniors'  goal,  then 

6  yards,  then  —  8  yards  {i.e.  it      ^ 

went   back   8   yards),  and    so 

on,  as  shown  in  the  diagram.     What  was  the  position  of  the 

ball  after  3  plays  ?  after  4  plays  ?  after  5  plays  ?  after  6  plays  ? 

20.  Plot  the  following  and  find  the  last  position  of  the  ball : 
On  15-yard  line ;  gained  4  yards;  gained  5  yards;  lost  2  yards; 

gained  30  yards  ;  lost  6  yards  ;  lost  2  yards  ;  gained  12  yards. 

21.  How  far  from  port  is  a  vessel,  if  it  sails  50  miles,  —  10 
miles  (driven  back  10  miles),  40  miles,  —  30  miles,  and  80  miles? 


45 


positivp:  and  negative  numbers  53 

62.  By  doing  the  work  in  §§  17-19,  the  student  has  become 
familiar  with  adding  literal  expressions  in  which  the  terms  are 
all  positive.  When  some  of  the  terms  are  negative,  the  method 
is  essentially  the  same,  the  only  difference  being  in  the  matter 
of  signs,  which  has  just  been  explained. 

EXERCISES 


63. 

Add: 

1. 

2x 
3x 

2.      a 

5a 

3.    -a 
4.a 

4.    -4c 
-3c 

5. 

4:V 

-2v 

-7v 

6.    -y 

4.y 

-9y 

7.      12  m6 
-2mb 
-6mb 

8.    40  a:* 
-lOar^ 
-60  a^ 

9. 

6a- 
2a- 
-5a 

-2b 
■f  36 
-46 

10. 

Sxy 
-2xy 
,    'Txy 

+  22/*         11. 
+  62/* 
-42/* 

10  x-{- 3  y-\-z 
-X-    y 

2x  +  2y  +  z 

12.  Simplify  11  a^b  -  1  ab'^  +  2  ac' +  10  ab  +  a(?  -  2  a^b -\' b^ 
+  5a62-2  63  +  2a6*-8a6-6a26. 

PROCESS 

lla-6-7a6*  +  2ac2  +  10o6+    b^ 
-2a*6  +  5a6*+    a(?  -2W 

-6a*6  +  2a6*  -    8a6 

3a26  ^3acr+    2ab-    b^ 

B-ULE. — Arrange  the  terms  so  that  similar  terms  shall  stand  in 
the  same  column. 

Find  the  algebraic  sum  of  each  column,  and  write  the  results  in 
succession  ivith  their  proper  signs. 

13.  Simplify  2a  +  26  +  3c  +  46-4a  +  6a-2c. 

14.  Simplify  lw-\-'lx  — A:y  —  x  —  2ic-\-3y  —  3x-\-AiW. 

15.  Simplify  7Z-6m  +  3w  —  8Z  +  4m  +  ll?i+m— 2?. 

16.  Simplify  15r-\-6s-llt-\-r-9 s+t-2 s-{-5r-2t-10r. 


54  POSITIVE   AND  NEGATIVE  NUMBERS 

Simplify  the  following  polynomials  : 

17.  7 x-lly-{-4:Z-7  z-{-llx-4:y-\-7  y-llz-4:X-\-y-x-z. 

18.  a  +  36  +  5c-6a  +  d  +  46-2c-2&+5a-d  +  a-6. 

19.  4  x'-S  xy-\-5  f+ 10 xy- 17 y^-11  x'-5xy-\-12  a^-2  xy. 

20.  2xy-5y^-\-xy-7xy-\-3y''-4:i>^y^-{-5xy-\-4.y^-\-:ihf. 

21.  Add2a-3&,  2b -3c,  5  c-Aa,  10a-5h,  and76-3c. 

22.  Add  x  +  y-\-z,  X  —  y-i-z,  y  —  z  —  X,  z  —  X  —  y,  and  x  —  z. 

23.  Add  4:a^-2x'-7x-\-l,  a^-{-3  3(^-\- 5  X- 6,4:  x^-Sa^-\-2 
-6x,2s(^-2x^  +  Sx-\-4:,2ind2x^-Sx'-2x-{-l, 

24.  Add5a;-32/-2;2,  4i/-2a;+6  0,3a-2a;-4  2/,  4  6-22 
—  5y,  a—  5b,  5y  —  6x,  Sx-{-2y  —  5a  —  2b,  and  6x  —  y 
-2  2  +  4  6. 

25.  Add  .12a.'3-4a^4-aJ  +  2,  .4^72  -  4  a?  +  .  4  -  ic^,  3ia;-.6 
4- 3  a.-^ -f  2  a.-^,  and  1  -  i  a?  + 1 . 2  a;'^  +  II  a^. 

26.  Add  20  aj^"  —  4  a;"'?/'*  +  36  2/2%  4  a^'"?/'*  — 15  ?/2«  —  12  a;-'", 
3  /"  +  3  af"^,  4  a;"'?/*  —  11  a;^"'  —  16  y-"",  and  a.--"*  —  2/-'*. 

Difference  of  Two  Numbers 

EXERCISES 

64.  On  account  of  the  extension  of  the  scale  of  numbers 
below  zero  (§  51),  subtraction  is  always  possible  in  algebra. 

When  the  subtrahend  is  positive,  algebraic  subtraction  is  like 
arithmetical  subtraction,  and  consists  in  counting  backtoard 
along  the  scale  of  numbers,  as  illustrated  in  §  58. 

Subtract  the  lower  number  from  the  upper  one : 
1.       6  6  6  6  6  6  6 

3  4  5  6  7  8  9 


2. 


Observe  that  subtracting  a  positive  number  is  equivalent  to 
adding  a  numerically  equal  negative  number. 


-3 

-3 

-3 

-4 

-5 

-6 

-7 

0 

1 

2 

3 

4 

5 

6 

POSITIVE   AND   NEGATIVE  NUMBERS  55 

When  the  subtrahend  is  negative,  it  is  no  longer  possible  to 
subtract  as  in  arithmetic  by  counting  backward. 
3.    Subtract  —  2  from  8. 

Explanation.  —  If  0  wepe  subtracted  from  8,  the  result 
would  be  8,  the  minueBd  itself. 
8  The  subtrahend,  however,  is  not  0,  but  is  a  number  2 

—  2      units  below  0  in  tlie  scale  of  numbers.     Hence,  the  differ- 
8  _j_  2  =  10      6nce  is  not  8,  but  is  8  +  2,  or  the  minuend  plus  the  sub- 
trahend with  its  sign  changed. 

Or,  —  2  is  subtracted  from  8  by  beginning  at  8  in  the  scale  of  numbers 
and  counting  2  units  in  the  direction  opposite  to  that  indicated  by  the 
sign  of  the  subtrahend,  arriving  at  10. 

Remark. — Notice  that  any  number  is  added  by  counting  along  the 
scale  of  numbers  in  the  direction  indicated  by  its  sign;  and  any  number 
is  subtracted  by  counting  in  the  direction  opposite  to  that  indicated  by 
its  sign. 

Subtract  the  lower  number  from  the  upp6r  one: 

4.  4444579 
0       -1       -2       -3      z:±      zlL      nl. 

5.  _5        _5        _5         _5        -1         -4        -6 

0         -1^        -2         -6^        -3         -1_         -^ 

Observe  that : 

Principle. — Subtracting  any  number  {imsitive  or  negative)  is 
equivalent  to  adding  it  with  its  sign  changed. 

Subtract  the  lower  number  from  the  upper  one : 

6.  10  7.      12  8.      20  9.      16         10.      40 


11. 


16.        4         17.        4         18.    —4         19.    -9         20.    —  7 
4-4488 


—  2 

5 

-6 

-4 

-8 

0 

12. 

-3 

13. 

-7 

14. 

10 

15. 

-5 

-2 

-6 

4 

-5 

10 

56 


POSITIVE   AND   NEGATIVE   NUMBERS 


21.  Subtract  12  from  —1.  23.    From  0  subtract  —3. 

22.  Subtract  -  4  from  14.  24.    From  —  3  subtract  0 . 

25.  From  0  subtract  —7;    from  the  result  subtract   —4; 
then  add  —  2  ;  add  —  3 ;  add  7 ;  subtract  11 ;  and  add  —  6. 

26.  AVhich  is  greater  and  how  much,  3  or   —  5  ?     —  2  or 
-5?     6or8-3?     _  2  + (-8)  or  -2  -  ( -8)? 

A  weather  map  for  January  16  gave  the  following  minimum 
and  maximum  temperatures  (Fahrenheit)  : 


Chicago 

DXTLUTH 

Helena 

MONTBEAL 

New  Orleans 

New  York 

Minimum 
Maximum 

24° 
30'' 

-6° 

2° 

-12° 

-4° 

-12° 
18° 

64° 
76° 

20° 
42° 

27.  The  range  of  temperature  in  Chicago  was  6°.  Find  the 
range  of  temperature  in  each  of  the  other  cities. 

28.  The  freezing  point  is  32°  F.  How  far  below  the  freez- 
ing point  did  the  temperature  fall  in  Montreal  ? 

29.  How  much  colder  was  it  in  Duluth  than  in  Chicago  ? 
in  Montreal  than  in  New  York?  in  Helena  than  in  New 
Orleans  ? 

30.  An  elevator  runs  from  a  basement,  —  22  feet  above  the 
first  floor,  to  the  tenth  story,  105  feet  above  the  first  floor. 
Express  its  total  rise  from  the  basement  to  the  tenth  floor; 
from  the  tenth  floor  to  the  basement. 

65.  From  the  work  of  this  chapter,  the  student  will  have 
discovered  that  negative  numbers  give  the  definitions  of  addi- 
tion, subtraction,  sum,  and  difference  a  wider  range  of  mean- 
ing than  they  had  in  arithmetic.  In  algebra  addition  does  not 
always  imply  an  increase,  nor  subtraction  a  decrease. 

In  §§  20,  21,  the  student  learned  how  to  subtract  one  literal 
expression  from  another,  all  the  terms  being  positive  and  the 
subtrahend  being  less  than  the  minuend.  This  is  arithmetical 
subtraction.  He  will  now  apply  the  broader  algebraic  idea  of 
subtraction  to  literal  expressions. 


POSITIVE   AND   NEGATIVE  NUMBERS  57 

EXERCISES 

66.    1.   From  10  x  subtract  15  x. 

PROCESS 

^^^  Explanation.  —  Since  (§  64,  Prin.)  subtracting  any 

15  X  number  is  equivalent  to  adding  it  with  its  sign  changed, 

—  15  ic  may  be  subtracted  from  10  x  by  changing  the  sign 

H    5^  of  15  X  and  adding  —  15  x  to  10  x. 

2.  3. 

From       5  a  5  a; 

Take       2_a        -2x 

7.   From  8  a;  —  3  y  subtract  5x  —  7y. 

PROCESS  Explanation. —Since  (§  64,  Prin.)  subtracting 

Sx  —  Sy  any  number  is  equivalent  to  adding  it  with  its  sign 

5  a;  —  7  y  changed,  subtracting  5  x  from  8  a:  is  equivalent  to 

I  adding  —  5x  to  8a-,  and  subtracting  —  7  y  from 

Q       I  j^  —  Sy  ia  equivalent  to  adding  +  7  y  to  —By. 

Rule.  —  Change  the  sign  of  each  term  of  the  subtraJiend,  and 
add  the  result  to  the  minuend. 

After  a  little  practice  the  student  should  make  the  change  of  sign 
mentally. 


4. 

5. 

6. 

9  am 

—  Smn 

3a,V 

21  am 

—  4m?i 

-lOx'y' 

8. 

9. 

10. 

11. 

From 

9a  +  7b 

5r-10s 

10x-\-2y 

3m-3n 

Take 

2a  +  36 

7r+   4s 

6x-4ty 

2m-5n 

12.  From  5x  —  3y-{-z  take  2x  —  y  +  Sz. 

13.  From  3  a^b  -j-b^-a^  take  i  a^b  -  S  a^ -{- 2  b^ 

14.  From  13a2  +  5&2_4c2  take  8a2  +  952^10c2. 

15.  From  15x  —  3y  +  2z  subtract  Sx-^Sy  —  9z. 

16.  From  a"-ab-  b^  subtract  ab-2a^-2  b\ 

17.  From  m^  —  mn  -f  n*  subtract  2m^—3  mn  +  2 n^. 

18.  From  4  a;^  _|_  3  ^^  ^  ^2  subtract  2  a;^  _  5  ^.^  ^  2  y^ 


58  POSITIVE   AND   NEGATIVE   NUMBERS 

19.  From  S  ab-\-a^-\-b^  subtract  a^  +  4  a6  +  61 

20.  From  6  oc^  -{- 4:  xy  —  3  i/  subtract  4:y-  —  3xy  -{-6  ^. 

21.  From  the  sum  of  3  a^  —  2  a6  —  6^  and  3  a6  —  2  a^  subtract 
a^—  ah  —  b'\ 

22.  From  3x  —  y-\-z  subtract  the  sum  of  x  —  4:y-\-z  and 
2x  +  Sy-2z. 

23.  From  a-{-b  +  c  subtract  the  sum  of  a—b  —  c,  b  —  c  —  a, 
and  G  —  a  —  b. 

24.  Subtract  the  sum  of  m^n  —  2mn^  and  2m~u  —  7n^  —  7i^ 
+  2  mn^  from  m^  —  n^. 

25.  Subtract  the  sum  of  2c  — 9a  — 36  and  36  — 5a  — 5c 
from  6  —  3  c  +  a. 

26.  From  the  sum  of  3  0?"*+ 4?/'* +2;"'+"  and  2  z'^+'' -\- 2x^^—3  y'' 
subtract  4  ic"*  —  2  2/"  +  z'^+\ 

Ux  =  a^-hb\y  =  2ab,z  =  a'-  b\  and  v  =  a^  -  2  a6  +  6^, 

27.  a?  +  ?/H-2;  +  v  =  ?  29.    a;  — y  +  s;  — 'y=? 

28.  x  —  y  —  z-\-7^  =  l  30.    2/  — ^— '^  +  2;  =  ? 

TRANSPOSITION  IN  EQUATIONS 

67.  In  the  solution  of  equations  the  student  has  used  certain 
principles,  stated  in  §  40  and  §  42  and  summed  up  in  §  44. 

They  are  usually  stated  in  somewhat  broader  terms  as  in 
the  following  section  and  are  so  simple  as  to  be  self-evident. 
Such  self-evident  principles  are  called  axioms. 

68.  Axioms.  —  1.  If  equals  are  added  to  equals,  the  sums  are 
equal 

2.  If  equals  are  subtracted  from  equals,  the  remainders  are 
equal. 

3.  If  equals  are  midtipUed  by  equals,  the  products  are  equal. 

4.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

In  the  application  of  axiom  4,  it  is  not  allowable  to  divide  by  zero  or 
any  number  equal  to  zero,  because  the  result  cannot  be  determined. 


POSITIVE   AND  NEGATIVE  NUMBERS  59 

EXERCISES 

69.  1.    Solve  a;  —  6  =  4  by  adding  6  to  both  members  (Ax.  1). 

2.  Solve  the  equation  a;  -f-  3  =  11  by  employing  Ax.  2. 

3.  Solve  I  ic  =  10  by  employing  Ax.  3. 

4.  Solve  7  a;  =  21.     Explain  how  Ax.  4  applies. 

5.  Solve  !»=  16  in  two  steps,  first  finding  the  value  of  ^x 
by  Ax.  4,  then  the  value  of  x  by  Ax.  3. 

Solve,  and  give  the  axiom  applying  to  each  step : 

6.  2.^  =  6.  13.    ic-h2  =  10.  20.   fm  =  9. 

7.  5x  =  5.  14.    IV  — 5  =  11.  21.    1 71  =  8. 

8.  4?/ =  8.  15.    i(;-}-l  =  12.  22.    fa;  =  10. 

9.  3?/  =  9.  16.    .s-7  =  10.  23.   fa;  =  21. 

10.  iz  =  5.  17.   9  +  .s  =  12.  24.    |z  =  20. 

11.  iz  =  2.  18.    r)-\-y  =  lo.  25.    |«  =  15. 

12.  \v  =  S.  19.    10  +  .y  =  12.  26.    iw  =  49. 

70.  1.    Adding  7  to  both  members  of  the  equation 

a-  -  7  =  3, 
we  obtain,  by  Ax.  1,  a;  =  3  +  7. 

•    —  7  has  been  removed  from  the  first  member,  but  reappears 
in  the  second  member  with  the  opposite  sign. 

2.  Subtracting  5  from  both  members  of  the  equation 

X  +  T)  =  9, 
we  obtain,  by  Ax.  2,  .«  =  9  —  5. 

When  plus  5  is  removed,  or  transposed,  from  the  first  mem- 
ber to  the  second,  its  sign  is  changed. 

3.  Explain    the    transposition    of    terms   in   each   of    the 
following  : 


2a;-l=5; 
2a;  =  5  +  l. 


3.^•  +  2  =  ll; 
3a;=ll-2. 


4a;  =  14-3a; 
4  a;  4-  3a;  =  14. 


71.   Principle.  —  Any  term   may  be  transposed  from    one 
member  of  an  equation  to  the  other,  provided  its  sign  is  changed. 


60  POSITIVE   AND  NEGATIVE   NUMBERS 

EXERCISES 

72.    1.  Solve  the  equation  6  —  5  ic  + 18  =  6  -f  3  a;  —  30. 

Solution 
By  Ax.  2,  ^- 5*- +  18  =  ^  +  3a;-30. 

Transposing,  §  7 1,  -  5  a;  -  3  x  =  -  30  -  18. 

Uniting  terms,  —  8  a;  =  —  48. 

Changing  signs,  8  x  =  48. 

Dividing  by  8,  Ax.  4,  x  =  6. 

Verification.  —  Substituting  6  for  x  in  the  given  equation, 

6  _  5  .  6  +  18  =  6  +  3  •  6  -  30,  or  -  6  =  -  G. 
Hence,  6  is  the  true  value  of  x  ;  that  is,  the  value  6  substituted  for  x 
satisfies  the  equation. 

Suggestions. — 1.  By  Ax.  2  the  same  number  may  be  subtracted,  or 
canceled,  from  botli  members. 

2.  By  Ax.  2  the  signs  of  all  the  terms  of  an  equation  may  be  changed, 
for  each  member  may  be  subtracted  from  the  corresponding  member  of 
the  equation  0  =  0. 

Solve  and  verify : 

2.  3  =  5-0;.     .  10.  8  +  7a  =  5a+20. 

3.  9-5a;==-l.  11.  2  +  13/^  =  50-9. 

4.  10  +  ^  =  18-^.  12.  50-n  =  20  +  n, 

5.  2z-\-2  =  S2-z.  13.  3x- 23  =  a; -17. 

6.  7x-\-2  =  x-\-U.  14.  4a;  +  12  =  2a;  +  36. 

7.  3p  +  2=p  +  30.  15.  2a;  +  Ja;  =  30-|-a;. 

8.  6y-2  =  3y  +  7.  16.  Sx-^x^SO  +  lx. 

9.  5  m  — 5  =  2m-f  4.  17.  5iv —  10  =  ^w  +  16. 

Simplify  as  much  as  possible  before  transposing  terms,  solve, 
and  verify : 

18.  10  a; +  30 -4  a; -9  a;  4- S3 +  12  a;  =  90 +12 -4  a;. 

19.  16a;  +  12-75  +  2a;-12-70  =  8a;-50-25. 

20.  lls-60  +  5s  +  17-2s  +  41-3s  =  2s  +  97. 

21.  10  2  -  35  - 12  2  + 16  +  32  =  4  ;2  -  35  + 10  ;2  +  32. 

22.  14  w  -  25  =  19  -  11  n  +  4  +  16  -  10  n  +  w  + 136  -  16  n. 


POSITIVE   AND  NEGATIVE   NUMBERS  61 

Algebraic   Representation 

73.    1.   How  much  does  8  exceed  S-\-2?  z  exceed  10 +  y? 

2.  What  number  must  be  added  to  5  so  that  the  sum  shall 
be  9  ?  to  m  so  that  the  sum  shall  be  4  ? 

3.  George  rode  a  miles  on  his  bicycle ;  then  b  miles  on  the 
cars;  and  walked  3  miles.     How  far  did  he  travel  ? 

4.  A  man  bought  a  house  for  m  dollars ;  spent  n  dollars 
for  improvements ;  and  then  sold  it  for  s  dollars  less  than  the 
entire  cost.     How  much  did  he  receive  for  it  ? 

5.  If  40  is  separated  into  two  parts,  one  of  which  is  Xy 
represent  the  other  part. 

6.  A  man  made  three  purchases  of  a,  b,  and  2  dollars, 
respectively,  and  tendered  a  20-dollar  bill.  Express  the  num- 
ber of  dollars  in  change  due  him. 

7.  Represent  three  times  a  number  plus  five  times  the 
double  of  the  number. 

8.  What  two  integers  are  nearest  to  8  ?  to  a;,  if  x  is  an 
integer  ?  to  a  -j-  6,  if  a  +  6  is  an  integer  ? 

9.  What  are  the  two  even  numbers  nearest  to  8  ?  What 
are  the  two  even  numbers  nearest  to  the  even  number  2n? 

10.  Express  the  two  odd  numbers  nearest  to  the  odd  num- 
ber 2  ?i  +  1 ;  the  two  even  numbers  nearest  to  2  n  -f- 1. 

11.  There  is  a  family  of  three  children,  each  of  whom  is  2 
years  older  than  the  next  younger.  When  the  youngest  is  x 
years  old,  what  are  the  ages  of  the' others?  When  the  oldest 
is  ?/  years  old,  what  are  the  ages  of  the  others  ? 

12.  A  boy  who  had  2  dollars  spent  25  cents  of  his  money. 
How  much  money  had  he  left  ?  If  he  had  x  dollars  and  spent 
y  cents  of  his  money,  how  much  money  had  he  left  ? 

13.  The  number  876  may  be  written  300  +  70+6.  Write 
the  number  whose  first  digit  is  x,  second  digit  2/j  and  third 
digit  z. 


62  POSITIVE   AND  NEGATIVE  NUMBERS 

Problems 

74.  If  3  a;  =  a  certain  number  and  a;  +  10  =  the  same  num- 
ber, then,  3ic  =  a;  +  10. 

This  illustrates  another  axiom  to  be  added  to  the  list  that  is 
given  in  §  68. 

Axiom  5.  —  N^tmbers  that  are  equal  to  the  same  number ,  or  to 
equal  numbers,  are  equal  to  each  other. 

This  axiom  is  useful  in  the  solution  of  problems,  for  its  application  is 
always  involved  in  writing  the  equation  of  the  problem. 

76.  The  student  is  advised  to  review  the  general  directions 
for  solving  problems  given  on  page  45. 

1.  The  Borough  of  Manhattan  contains  22,000  elevators.  If 
2000  more  are  for  freight  than  for  passengers,  how  many 
freight  elevators  are  there  ? 

2.  The  total  height  of  a  certain  brick  chimney  in  St.  Louis 
is  172  feet.  Its  height  above  ground  is  2  feet  more  than  16 
times  its  depth  below.     How  high  is  the  part  above  ground  ? 

3.  There  are  3141  of  the  Philippine  Islands,  of  which  the 
number  that  has  been  named  is  195  more  than  the  number 
that  is  nameless.     Find  the  number  of  each. 

4.  The  value  of  the  toys  made  in  Germany  one  year  was 
^22,500,000,  or  $100  more  than  4  times  the  amount  purchased 
by  the  United  States.     Find  the  value  of  the  latter's  purchase. 

5.  The  Canadian  Falls  in  the  Niagara  Eiver  are  158  feet 
high.  This  is  8  feet  more  than  \^  of  the  height  of  the  Ameri- 
can Falls.     Find  the  height  of  the  American  Falls. 

6.  The  summer  bird  population  of  Illinois  is  estimated  at 
30,750,000. and  the  number  of  English  sparrows  is  19,750,000 
less  than  the  number  of  other  birds.  Find  the  number  of 
sparrows. 

7.  The  porch  of  a  temple. in  India  is  876  feet  in  perimeter, 
and  \  of  its  length  is  6  feet  more  than  its  width.  Find  its 
length  and  width. 


POSITIVE   AND   NEGATIVE   NUMBERS  63 

8.  With  the  machines  of  the  present  time  a  pin  maker  can 
turn  out  1,500,000  pins  a  day,  or  60,000  more  than  300  times 
the  daily  output  of  a  pin  maker  of  early  times.  How  many 
pins  did  the  early  pin  maker  turn  out  per  day  ? 

9.  The  cost  of  dressing  the  fur  of  a  beaver  is  2  cents  more 
than  8  times  that  for  a  muskrat.  For  a  muskrat  the  cost  is 
9  cents  less  than  for  a  mink.  If  the  cost  of  dressing  all  three 
furs  is  71  cents,  find  the  cost  of  dressing  a  beaver's  fur. 

10.  The  daily  consumption  of  water  per  person  in  New 
York  City  is  22  gallons  less  than  that  in  Boston.  The  daily 
consumption  in  Pittsburg  is  250  gallons,  or  30  gallons  less  than 
that  in  New  'York  and  Boston  together.  Find  the  daily  con- 
sumption per  person  in  Boston. 

11.  A  carpenter,  a  plumber,  and  a  mason  together  earn 
$12.70  a  day.  If  the  carpenter  earns  $1.70  less  than  the 
mason,  and  he  and  the  plumber  together  earn  $  7.50,  how  much 
does  each  earn? 

12.  A  letter  sent  from  Indianapolis  to  Point  Barrow,  Alaska, 
travels  6800  miles.  It  goes  900  miles  more  by  train  than  by 
steamer  and  200  miles  more  by  dog  sleds  than  by  train.  How 
far  does  it  travel  by  each  ? 

13.  In  the  first  three  years  of  excavation,  313,356  cubic  yards 
more  were  taken  from  the  Panama  Canal  than  from  the  New 
York  Barge  Canal.  The  amount  taken  from  the  former  was 
6,364,484  cubic  yards  less  than  twice  that  from  the  latter. 
Plow  much  was  excavated  from  each  ? 

14.  Of  the  wood  used  for  pulp  in  New  York  State  one  year, 
500,000  cords  were  supplied  by  the  state.  The  amount  im- 
ported was  f  of  that  used  by  Maine.  If  New  York  used  twice 
as  much  as  Maine,  how  many  cords  were  used  by  each  ? 

15..  At  one  time  the  coffee  stored  at  the  docks  of  Havre, 
France,  was  |  of  the  total  yearly  production  of  the  world  and 
^  that  of  Brazil.  If  Brazil  produces  3^  million  bags  more  than 
all  the  rest  of  the  world,  find  the  amount  stored  at  Havre. 


64  POSITIVE   AND   NEGATIVE  NUMBERS 

MULTIPLICATION 

76.  Primarily  multiplication  is  the  process  of  taking  one 
number  as  many  times  as  there  are  units  in  another. 

Thus,  3  X  5  =  5  +  5  +  5  =  15. 

In  this  section  and  the  next,  the  sign  x  is  to  be  read  '  times.' 

Even  in  arithmetic  multiplication  is  extended  to  cases  that 
cannot  by  any  stretch  of  language  be  brought  under  the  original 
definition. 

Thus,  strictly,  in  3|  x  4,  4  cannot  be  taken  3f  times  any  more  than  a 
revolver  can  be  fired  3|  times. 

So  in  algebra  there  are  still  other  cases,  like-— 3  x  4,  that 
do  not  come  under  the  original  definition. 

What  we  are  concerned  with,  however,  is  the  method  of  find- 
ing the  product  (consistent  with  the  laws  of  operation  used  in 
arithmetic)  and  the  interpretation  of  the  results  obtained. 

77.  Sign  of  the  product. 

1.  Just  as  in  arithmetic  3  times  4  are  12,  so  3  times  4  posi- 
tive units  are  12 positive  units; 

that  is,  +3x+4  =  +  12.  (1) 

2.  Also,  3  times  4  negative  units  are  12  negative  units ; 
that  is,  ■        +3x-4  =  -12.  (2) 

3.  Just  as  in  arithmetic  4x3  =  3x4, 
so  — 3x4=:4x— 3; 
and  since                            — 12  =  4  x  —  3, 

-3x4  =  -12.  (3) 

4.  Again,  since  6  —  4  =  2,  . 

by  Ax.  3,  -3(6-4)  =-3x2  = -6. 

Also,  §27,  -3(6-4)  =  -3x6-3x-4  =  -18-3x-4. 

Now,  since  both  — 18  — 3x— 4  and  —6  are  equal  to  —3(6—4), 
by  Ax.  5,  _i8-3x-4  =  -6. 

Transposing  —  18,  §  71, 

_3x-4  =  -6  +  18; 
that  is,  _3x -4  =  4-12.  (4) 


POSITIVE   AND   NEGATIVE   NUMBERS  6b 

5.  The  preceding  conclusions  may  be  written  as  follows: 

From  (1),  +  a  X  +  6  =  +  a&, 

from  (2),  -\-ax  —  b  =  -ab, 

from  (3),  —  a  X  +  &  =  —  a6, 

and  from  (4),  —  a  x  —  6  =  +  a6. 

Hence,  for  multiplication : 

78.  Law  of  signs. — Tfie  sign  of  the  product  of  two  factors  is  -\- 
when  the  factors  have  like  signs,  and  —  when  they  have  unlike 
signs. 

EXERCISES 

79.  1.  Multiply  each  of  the  following  by  +  2 ;  then  by  —  2 : 

3,  5,  -6,  10,   -8,  -9,  -12,  a,  x,  -b. 

2.  Multiply    -8  9  6  4-2 
By              _6               1            Zl5            nI  10 

3.  Multiply        a  —6  —x  —y  n 
By                  1            _J?.           jzl            Zll             -12 

80.  When  there  are  several  factors,  by  the  law  of  signs, 

—  a  X  —b  =  -{-ab', 
—  a  x—bx—c  =  -{-abx  —  c  =  —  abc ; 
—  ax  —  bx—cx—d  =  —  abc  x  —  d  =  +  abed ;  etc.     Hence, 
The  product  of  an  even  number  of  negative  factors  is  positive; 
of  an  odd  number  of  negative  factors,  negative. 

Positive  factors  do  not  affect  the  sign  of  the  product. 

EXERCISES 

81.  Find  the  products  indicated : 

'l.  (_1)(_1)(-1).  6.  (-!)(_ 2) (-3) (-4). 

2.  (-2)(-a)(-6)c.  7.  (_a)(-6)(+c)(-rf). 

3.  (-l)(x)(3,)(-36).  8.  (-x)(-y){-l)zi~v). 

4.  (-3)(2)(-2a)6.  9.  (-x)(y)(z)(-v)(-io). 

6.  (-2a){-3b)(-c)x.  10.    (-r)(-s)(-0(x)(-3y). 

milnk's  1st  yr.  alo. — 6 


66 


POSITIVE  AND  NEGATIVE   NUMBERS 


82.  To  multiply  when  the  numbers  are  either  positive  or  nega- 
tive. 

Having  learned  to  multiply  when  the  numbers  are  posi- 
tive, §§  22-29,  and  having  just  learned  about  the  sign  of  the 
product  when  there  are  negative  factors,  the  student  is  now 
prepared  to  multiply  whether  the  terms  of  the  factors  are  posi- 
tive or  negative. 

EXERCISES 

83.  1.   Multiply  -4  aar^  by  2  aV. 

Explanation.  —  Since  the  signs  of  the  monomials  are 
unlike,  the  sign  of  the  product  is  —    (Law  of  Signs, 
§78). 
4-2=8  (Law  of  Coefficients,  §  24). 
a .  a^  =  «! .  aS  =  «i+3  _  ^4  (Law  of  Exponents,  §  23). 
cc2 .  a;4  =  x2+4     =  0^  (Law  of  Exponents). 

Hence,  the  product  is  —  8  a^oifi. 


PROCESS 

—  4aa^ 

2aV 

-8aV 

Multiply : 
2.    10  a' 
5a« 


—  5  m^n^ 
Smn 


6.  xy 


10. 

-2x 

2a^ 

14. 

-Zn^ 

6  b' 

18. 

2a-+i 

3a' 

22. 

d*-« 

^2s+n 

7. 

-2rq^x 

11. 

-Ga'c'x 
-4.a'bn 

15. 

4.a'bY 
Sa'b'y 

19. 

-  2  a'-b'' 
Ian'" 

23. 

8  7--" 
3  r"s«-2^ 

4. 


8. 


12. 


—  4  abc 

2a^b 

-Sab 
-1 

-Sab 
2ba 


16.     — 


1 
-1 


5.        3  a^b(^ 

-  7  abH 

9.    -5aV 
~2ajx^ 

13.    -2a^y? 

-  4aa;* 

17.    -hm'^d' 

-  2  m'^'cd' 


20. 


x'Y 
xy 


21. 


4a;"-i 

-2a;"+i 


24.    -a^- 

—  a?" 


25.'  2/""" 

ym-n+l 


26.  Sx^-2xyhj  5xf. 

27.  3a'-6a'bhj  -2b. 

28.  TTv^n^  —  3  m/i'*  by  2  mn. 


29.  j^V— 2i)§^by  —  pg. 

30.  4  a^  -  5  ft^c  _  c^  by  aftc^. 

31.  —  2  ac -f  4  aa?  by —5  aca;. 


POSITIVE   AND   NEGATIVE  NUiMBEKS  67 

Expand,  and  test  each  result : 

32.  (a-f-&-fc)(a  +  6-f  c). 

33.  {a^-ah  +  y'){a?  +  ah  +  h-). 

34.  (a^  +  3a-6  +  3a62-t-63)(tH-6). 

35.  {a-h){a  +  h){a'-^h''){p>  +  b'). 

36.  (»^  —  ^y-\-  ^]f-  —  x\f-\-  y*)  (x  +  y). 

37.  (a2  +  62  +  c2  +  d2)(a2_62-|.c2-cF). 

38.  (x'-  xy-\-y'-{-x-{-y-^l)(x-{-y  +  l). 

39.  (a-^  +  3  a'b  -\-3ab^-\-  b^)  (a"  H-  2  a6  +  b^. 

40.  (a2_a6  — «c4-&^-&c  +  c2)(a  +  6-f  c). 

Arrange  both  multiplicand  and  multiplier  according  to  the 
ascending  or  the  descending  powers  of  the  letter  involved; 
multiply,  and  test  each  result. 

41.  x-\-a:^-{-l-\-x^hy  x—1. 

42.  a^  +  10-7x-4:X^hjx-2. 

43.  U-9x-6x^-\-x^hj  x-^1. 

44.  a»-30-lla  +  4a2by  a-1. 

45.  4a2_2a3-8a  +  a*-3by  2  +  a. 

46.  2m-3  +  2m3-4m2by  2m-3. 

47.  x  +  x^-5hj  x^-3-2x. 

48.  62 _^56-4by  -4  +  252-36. 

49.  4n8  +  6-2n*H-16n-87i2  +  n«by  n  +  2. 

50.  l  +  aJ4-4a^  +  10a^  +  46.r*  +  22ic<by  2a^-f-l-3a;. 

51.  5x^  +  7 -4:X^-6x-^x^-j-Sx'-2x^hy2x-{-x^-{-l. 

Multiply : 

52.  ax^**  +  a/«  by  aa:^"  -  ay^. 

53.  aa;«-i  +  2/'*"^  by  3  a.-c""^  +  2  i/"-\ 

54.  ic^n  ^  2  ;t;"?/"  +  2/2«  by  a^'.  _  2  af»2/"  +  2/^. 

55.  a^"  +  a^^b^  +  a^'^ft^^  +  6«^  by  a^"  -  6^^ 

56.  m*+V-^  -f-  ?/i*-^7i*+i  + 1  by  m'"  V+^— m^+^Ti'-^  +  1, 


68  POSITIVE   AND   NEGATIVE   NUMBERS 

Special  Cases  in  Multiplication 

84.  The  square  of  the  sum  of  two  numbers. 

1.  Multiply  a  +  6  by  a  -f  6 ;  find  the  square  of  x-\-y. 

.    a-{-b  x  +  y 

a  +  b  x-)ry 

a^  -\-ah  x^  +  xy 

2.  How  is  the  first  term  of  the  product,  that  is,  of  the  square 
of  the  sum  of  two  numbers,  obtained  from  the  numbers  ?  How 
is  the  second  term  obtained  ?  the  third  term  ? 

3.  What  signs  have  the  terms  of  the  result? 

85.  Principle.  —  Hie  square  of  the  sum.  of  two  numbers  is 
equal  to  the  square  of  the  first  number,  plus  twice  the  product  of 
the  first  and  second,  plus  the  square  of  the  second. 

Since  5a^x5a^  =  25  a%  3  a^b'  x  3  a*b'  =  9  a^b^"",  etc.,  it  is  evi- 
dent that  in  squaring  a  number  the  exponents  of  literal  factors 
are  doubled. 

EXERCISES 

86.  Expand  by  inspection,  and  test  each  result : 

1-  (i>  +  g)(i>  +  g).  11.  (505-1-2!/.  21.  (a^-^by. 

2.  (r-\-s)(r  +  s).  12.  (2a-{-xy.  22.  (a^  +  b^f. 

3.  (a-\-x)(a-{-x).  13.  (ab  +  cdf.  23.  (a'^  +  ft**)". 

4.  (a; +  4)  (a; +  4).  14.  (5x  +  2yy.  24.  (af«  +  ^«)2. 

5.  (a-h6)(a  +  6).  15.  (7z-^Sc)\  25.  (Sa'-^5by. 

6.  (2/  +  7)(y-f-7).  16.  (Sb  +  4:x)\  26.  (l-\-2a'by. 

7.  (z  +  l)(z  +  l).  17.  (2m-f-3n)2.  27.  {S  xy^  +  4:  a^yf. 

8.  (c  +  9)(c-{-9).  18.  (3c-f-7d!)2.  28.  (9  a-'"-f-2  62«)2. 
9  (v-\-S)(v-hS).  19.  (8  s  +  5  ty.  29.  (4  aj^r  4.  y^+iy 

10.   (w-\-o)(w-\-5).     20.  (5w-{-3uy.  30.  (a;-^  + 1/""^^. 


POSITIVE   AND  NEGATIVE  NUMBERS  69 

87.  The  square  of  the  difference  of  two  numbers. 

1.  Multiply  a  —  bhya  —  b',  find  the  square  of  x  —  y. 

a—b  x—y 

a—b  x—y 

a?—ab  a?  —  xy 

-ab-\-b^  —xy  +  f 

a'-2ab-^b^  x'-2xy-\-f 

2.  How  is  the  first  term  of  the  square  of  the  difference  of 
two  numbers  obtained  from  the  numbers  ?  How  is  the  second 
term  obtained  ?  the  third  term  ? 

3.  What  signs  have  the  terms  of  the  result  ? 

88.  Principle. —  The  square  of  the  difference  of  two  numbers 
is  equal  to  the  square  of  the  first  number,  minus  tivice  the  product 
of  the  first  and  second,  i)lus  the  square  of  the  second. 

EXERCISES 

89.  Expand  by  inspection,  and  test  each  result : 

1.  {x-m){x-m).  14.  {2a-xy.  27.  (3a;-2)«. 

2.  {m-n){m-n).  15.  {Sm-nf.  28.  {2x-5y)\ 

3.  (a;-6)(a;-6).  16.  {^x-y)\  29.  (5m-3w)*. 

4.  (p_8)(i)-8).  17.  (m-4n)2.  30.  {^p-bq)\ 

5.  (9-7)(g-7).  18.  {p-^qf.  31.  (a* -6"/. 

6.  (a  -  c)  (a  -  c) .  19.  (a  -  7  bf.  32.  (af  -  y^'f. ' 

7.  (a -a;) (a -a;).  20.  (4 a -3)2.  33.  {a^-2by. 

8.  (a;-l)(a;-l).  21.  (5  a; -4)1  34.  (f-6xy. 

9.  (6 -5)  (6 -5).  22.  (ab-Sy.  35.  (ab-2(^\ 

10.  (st-2)(st-2).  23.  (2a-Sby.  36.  (4:0(^-5yy. 

11.  (a: -4) (a; -4).  24.  (2z-7yy.  37.  (xf-^'^y^y. 

12.  (2 -3) (2! -3).  25.  (Sx-5yy.  38.  (mx^-ny^y. 

13.  («;-9)(w-9).  26.  (9^v  —  2vy.  39.  (af-^-t/**" V- 


70  POSITIVE   AND  NEGATIVE  NUMBERS 


90 

1.   The  square  of  any  polynomial. 

1. 

Find  the  square  of  a  +  &  -f-  c. 

a-{-b-\-c 
a-\-b-\-c 
a^+    ab-{-    ac 

+    ab 

i-b'-\- 

be 

+    ac 

+ 

bC-^(^ 

a" -\- 2 ab -^2 ac -^  b^  +  2bc -i-  c" 
That  is,  {a-\-b  +  cy  =  a^-\-b^-{-c^-{-2a.b  ^2ac  +  2bc. 

2.  Show  by  actual  multiplication  that 

=  a^-\-b^  +  c'-\-d^-\-2ab-{-2ac-]-2ad-]-2bc-\-2bd  +  2cd. 

3.  Similarly,  by  squaring  any  polynomial  by  multiplication, 
it  will  be  observed  that : 

91.  Principle.  —  The  square  of  a  polynomial  is  equal  to  the 
sum  of  the  squares  of  the  terms  and  twice  the  product  of  each  term 
by  each  term,  taken  separately,  that  follows  it. 

When  some  of  the  terms  are  negative,  some  of  the  double  products  will 
be  negative,  but  the  squares  will  always  be  positive.  For  example,  since 
(-  6)2  =  +  &2^  (a  -  &  +  c)2  =  a=2  +(_  6)2  +  c2  +  2  a(-  6)  +  2 ac  +  2(-  6)c 
=  a2  +  62  +  c2  -  2  a6  +  2  ac  -  2  6c. 

EXERCISES 

92.  Expand  by  inspection,  and  test  each  result : 

•       1.  (x  +  y-[-zy,  3.    (x-y-zf.  5.    (x-\-y-Szy. 

2.  (x  +  y-zy.  4.    (x-y-\-zy.  6.    (x-y-{-3zy. 

7.  (a-26  +  c)2.  13.    (Sx-2y  +  4.zy. 

8.  (2a-b-cy.  14.    (2a-5b  +  3cy. 

9.  (rs  +  st-rty.  15     (2m-47i-r)2. 

10.  (qb-pc-rdy.  16.    (12-2x  +  3yy. 

11.  (ax  —  by -\- czy.  17.    (a -^  m -\- b -{- ny. 

12.  (xy  —  Sc  —  aby.  18.    (a  —  m  +  b  —  7if. 


POSITIVE   AND   NEGATIVE   NUMBERS 


71 


93.   The  product  of  the  sum  and  difference  of  two  numbers. 
1.    Find  the  product  of  a  -|-  6  and  a  —  6 ;  of  x  —  y  and  x  +  y. 


a  —  b 

-ah-b^ 
a'  -b' 


x-y 

^■¥y 
x^  —  xy 


2.  How  are  the  terms  of  the  product  of  the  sum  and  differ- 
ence of  two  numbers  obtained  from  the  numbers  ? 

3.  What  sign  connects  the  terms  ? 

94.   Principle.  —  The  product  of  the  sum  and  difference  of 
\wo  numbers  is  equal  to  the  difference  of  their  squares. 


EXERCISES 


95. 

Expand  by  inspection 

1. 

(x-\-y)(x-y). 

2. 

ia  +  c)(a-c). 

3. 

(P-^Q)(P-Qy 

4. 

(p  +  5)(i)-5). 

5. 

(x-\-l){x-l). 

6. 

(a^  +  l)(^_l). 

7. 

(a^  +  l)(a^-l). 

8. 

(x'-l){x^  +  l). 

9. 

(a^-l)(x^  +  l). 

10. 

(x-  +  f)(af-f). 

11. 

(ab-\-5)(ab-5). 

12. 

(cd-\-dr)(cd-cP). 

13. 

(ab-c^(ab  +  c^. 

14. 

(^y  +  z^(4.y-z^. 

15. 

(Ix  —  5  m)  (^a;  +  5  m). 

and  test  each  result : 

16.  (2x  +  ^y){2x-^y). 

17.  (3m  +  4n)(3m-4n). 

18.  {12  +  xy){\2-xy). 

19.  {ab-^cd){ab-cd). 

20.  (3m2n-6)(3m2n-(-6). 

21.  (2a?+5y^(2o^-5f), 

22.  (3a^  +  2/)(3a^-2/). 

23.  (2  a^  4- 2  62)  (2  a^- 2  62). 

24.  {-5n-b){-5n  +  b). 

25.  {-x-2y){--x  +  2y). 

26.  (3  af'  +  7  y")  (3  a;«  -  7  y"). 

27.  (maf  +  2  2/'')  (maf  -  2  2/*). 

28.  (a''6"'-f  a'^6'*)  (cC^b"^-  a'^b''). 

29.  (xT-^  +  y""^^)  (xT - 1  —  2/"+^). 

30.  (5  a^b""  +  2  x')  (5  a%^  -  2  af ).  ) 


72  POSITIVE   AND  NEGATIVE  NUMBERS 

96.   The  product  of  two  binomials  that  have  a  common  term. 

Let  a;  +  a  and  a;  4-6  represent  any  two  binomials  having  a 
common  term,  x.     Multiplying  a;  -f-  a  by  a?  +  6, 

x-\-a 
x-\-h 


o(f-\-ax 

bx-i-ab 


x^+  (a  +  b)x-^ab 

97.  Principle.  —  The  product  of  tivo  binomials  having  a 
common  term  is  equal  to  the  sum  of :  the  square  of  the  common 
term,  the  product  of  the  sum  of  the  unlike  terms  and  the  cominon 
term,  and  the  product  of  the  unlike  terms. 

EXERCISES 

98.  1.  Expand  (x-\-  2)(a5-f  5)  and  test  the  result. 

Solution 
The  square  of  the  common  term  is  x^  ; 
the  sum  of  2  and  5  is  7  ; 
the  product  of  2  and  6  is  10  ; 
.-.  (a;  +  2)(x  +  5)  =  a;2  +  7a;4-10. 
Test.  —  If  x  =  1,  we  have  3  •  6  =  1  +  7  +  10,  or  18  =  18. 

2.  Expand  (a  + 1)  (a  —  4)  and  test  the  result. 

Solution 
The  square  of  the  common  term  is  a^ ; 
the  sum  of  1  and  —  4  is  —  3  ; 

the  product  of  1  and  —  4  is  —  4 ;  •     ' 

.-.  (a  +  l)(a-4)  =  a2-3a-4. 
Test.  —  If  a  =  4,  we  have  5  •  0  =  16  -  12  -  4,  or  0  =  0. 

3.  Expand  (n  —  2)(n  —  3)  and  test  the  result. 

Solution 
The  square  of  the  common  term  is  n^  ; 
the  sum  of  —  2  and  —  3  is  —  5  ; 
the  product  of  —  2  and  —  3  is  6  ; 
.-.  (n-2)(ri-3)  =  n2-5n  +  6. 
Test.  —  If  n  =  3,  we  have  1  .  0  =  9  —  15  +  6,  or  0  =  0. 


POSITIVE   AND   NEGATIVE   NUMBEKIS  73 

Expand  by  inspection,  and  test  each  result : 

4.  (x-{-5)(x  +  6).  18.  (af'-5)(af  +  4). 

5.  (a;  +  7)(a;-f-8).  19.  (af»-a)(af -2  a). 

6.  (x-7){x-{-S).  20.  (y-2b){y-\-Sb). 

7.  (x-{-7)(x-S).  21.  (z-4:a)(z  +  3a). 

8.  (aj-5)(a;-4).  22.  (2  x-{-5)(2  x  +  S). 

9.  (a;-3)(a;-2).  23.  (2  a;- 7)  (2  a; +  5). 

10.  (a;-5)(a;-l).  24.  (3  y-lXSy-^2).        ^ 

.11.  (a;H-5)(a;  +  8).  25.  (-ia^ -\-l)(-ia^ -7), 

12.  (29-4)(/)  +  l).  26.  (a6-6)(a6  +  4). 

13.  (r-3)(r-l).  27.  (o^^- a)(a:2/  +  2  a). 

14.  (n-S)(n-12).  28.  (3xy -hf)(y'-xy). 

15.  (ri-6)(n-hl5).  29.  {h'^c? +  ef)(h^(? -ef). 

16.  (a^  +  5)(a:2-3).  30.  (5  ce&  +  2c2)(5a6  -  2  c^). 

17.  (a;3-7)(x3  +  6).  31.  (3  ar»  +  2  2/=^(3ar^- 2  1^^. 

By  an  extension  of  the  method  given  above,  the  product  of 
any  two  binomials  having  similar  terms  may  be  written. 

•  32.   Expand  (2  a;  -  5)  (3  a;  +  4) . 


PROCESS  Explanation.  —  The  product  must  have  a  term 

2  a;  —  5  in  r^,  a  term  in  x,  and  a  numerical,  or  absolute,  term. 

XThe  x*-term  is  the  product  of  2  x  and  3 x;  the  x-term 
is  the  sum  of  the  partial  products  —  5  •  3  x  and  2  x  •  4, 

*^  '^  "*"    called  the  cross-products ;  and  the  absolute  term  is  the 

6  ar^  —  7  a;  —  20     product  of  -  5  and  4. 

The  process  should  not  be  used  except  as  an  aid  in  explanation. 

Expand  by  inspection,  and  test  each  result : 

33.  (2a^4-5)(3a;  +  4).  38.    (2dH-5  6)(5  a  +  2  6). 

34.  (3a;-2)(2a;-3).  39.    (J  n' -2p)(2n^ -7  p). 

35.  (3a-4)(4a  +  3).  40.    (aft^  _  m^  (aft^  _  4  m^. 

36.  {3x-y)(x--3y),  41.    (4  ?•"•  -  3  s")  (2  ^-^  -  5  s**). 

37.  (72-a)(32  +  2a).  42.   (a^-i  +  5y)(2a^-i-3y). 


74  POSITIVE   AND  NEGATIVE  NUMBERS 

SIMULTANEOUS   EQUATIONS 

99.  If  4  bananas  and  9  oranges  cost  35  ^,  and  4  bananas  and 
6  oranges  cost  26  ^,  and  it  is  required  to  find  the  cost  of  1  of 
each,  we  may  simplify  the  problem  thus : 

4  bananas  and  9  oranges  cost  35  ^  (1) 

4  bananas  and  6  oranges  cost  26  ^  (2) 

Subtracting,  3  oranges  cost    9  p  (3) 

By  thus  eliminating  the  cost  of  the  bananas,  we  have  obtained 
a  relation,  (3),  more  simple  than  either  of  the  two  given  rela- 
tions, (1)  and  (2),  for  it  involves  only  one  unknown  cost. 

Or,  let  X  represent  the  number  of  cents  1  banana  costs,  and 
y  the  number  of  cents  1  orange  costs. 

Then,  4  bananas  will  cost  4  x  cents,  9  oranges  9  y  cents,  etc. 
4a;  +  9?/  =  35  (1) 

4a;  +  6y  =  26  (2) 

Eliminating  the  a;'s,  3y=    9  (3) 

y=    3,  or  1  orange  costs  3  ^. 

Since  y  =  S,9y  in  the  first  equation  is  equal  to  27. 
Substituting  3  for  y  in  the  first  equation, 

4  a;  +  27  =  35,  (4) 

x=    2,  or  1  banana  costs  2  ^. 

100.  In  eliminating  the  aj's  in  the  preceding  section,  equal 
numbers,  4  a;  +  6  i/  and  26,  were  subtracted  from  the  members 
of  (1).  Hence,  Ax.  2,  the  results  are  equal,  giving  a  true  equa- 
tion, 3  2/  =  9. 

This  method  of  elimination  is  called  elimination  by  subtrac- 
tion. 

101.  How  must  the  equations  2x-^3y=16  and  5x—S  y—19 
be  combined  to  eliminate  the  y^s  ? 

2x  +  3y  =  16 

5a;-3y  =  19 
Adding,  Ax.  1,  7  a;  =  35 

This  method,  of  elimination  is  called  elimination  by  addition. 


POSITIVE   AND   NEGATIVE  NUMBERS 


75 


102.  Equations  like  those  discussed  in  §  99  or  in  §  101,  in 
which  the  same  unknown  numbers  have  the  same  values,  are 
called  simultaneous  equations. 

EXERCISES 

103.  1.  If  2x-{-3y  =  lS  and  2  x  +  2/  =  10,  what  is  the  value 
of  each  unknown  number  ? 

Solution 
2  a;  +  3  y  =  18  (1) 

2x+    y  =  lO  (2) 

Subtracting,  Ax.  2,  2y=   8;  :.y  =  i. 

Substituting  4  for  2/  in  (2),    2  a:  +  4  =  10;  :.x  =  3. 

Test.  —  Substituting  3  for  x  and  4  for  y  in  (1)  and  (2), 

(1)  becomes  6  +  12  =  18,  or  18  =  18  ; 

(2)  becomes  6  +   4  =  10,  or  10  =  10. 

Note.  — The  value  of  y  may  be  substituted  in  either  of  the  given  equa- 
tions. 

Solve,  and  test  results: 
=  22, 
21. 


3. 


6. 


7. 


8. 


9. 


(5x-{-2y  = 
[5x-{-    y  = 

\3x-\-2y  =  19. 
r4a;  +  52/  =  32, 
[2x-\-5y  =  26. 
(7  x-2y  =  22, 
\Sx-\-2y  =  lS. 
i6x  +  7y  =  lS, 
\6x-\-  y  =  7. 
I9x-2y  =  n, 
[7x-2y^Sl. 
'5x-^Sy  =  16, 
2  a;  +  3  2/ =  10. 
(6x  +  2y=:22, 
[6x-7y  =  4:. 


10. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


Sx-4:y  =  16, 
5  a;  4-  4  y  =  48. 
(6x-^5y  =  70, 
1    x  —  5y  =  0. 

'  5  »  -  4  ?/  =  8, 
3  ic  —  4  2/  =  0. 

(Sx-^5y  =  lS, 

[Sx-\-8y  =  14:. 
7x-3y  =  39, 
5x-3y  =  27. 
5  y  —  4  a;  =  9, 

.  6  2/  -h  4  a;  =  46. 

(Sx-3y  =  39, 


Sx 


36. 


42/ 

I  7  a;  +  5  y  =  83, 
l7a;-42/  =  47. 


76 


POSITIVE   AND  NEGATIVE  NUMBERS 


18.   If  2  ic  -f-  3  2/  =  16  and  5  a;  -f  4  ?/  =  33,  find  x  and  y. 
Solution 

2x  +  Sy  =  lQ,  (1) 

6a;  +  4y  =  33.  '  (2) 

We  may  eliminate  either  x  or  y.  If  we  choose  to  eliminate  x,  we  must 
first  prepare  the  equations,  so  that  x  may  have  the  same  coeflBcient  in 
each.  Multiplying  both  members  of  (1)  by  5,  and  both  members  of  (2) 
by  2, 

10  a:  +  15  y  =  80  (3) 

and  10  a;  +   8^  =  66  (4) 

Subtracting  (4)  from  (3),  7y  =  U;  /.y  =  2. 

Substituting  2  for  ?/  in  (1),      2  a;  +  6  =  16  ;  ..x  =  6. 

Test.  —These  values,  in  (1)  and  (2),  give  10  +  6  =  16  and  25  +  8  =  33. 

Note.  —  To  eliminate  y  instead  of  x,  proceed  as  follows  : 

Multiplying  (1)  by  4,  Sx  +  12y  =  64. 

Multiplying  (2)  by  3,         16x-}-12y  =  99. 

Subtracting  the  upper  equation  from  the  lower,  thus  avoiding  negative 
coeflBcients, 

7  ic  =  35 ;  ..  a;  =  5. 
10  +  3!/  =  16;  .'.y  =  2. 


Substituting  5  for  a:  in  (1), 
Solve,  and  test  results : 
19. 


20. 


21. 


22. 


23. 


24. 


25. 


(9x-\-2y  =  20, 

[Sx+    y  =  l. 

r6a;  +  52/  =  28, 

|2aj  +  32/  =  12. 

|aj-2/  =  2. 
'5a;  +  22/  =  49, 

3a;-22/  =  23. 

4  a; -2/ =  27, 
x  -  2/  =  3. 

{2x+    2/  =  13, 
\     a;  +  42/  =  17. 

r42/-3a:  =  30, 
l52/-6£c  =  33. 


26. 


27. 


28. 


29. 


30. 


31. 


riO.'B  +  32/  =  62, 
I    6a;  +  4?/  =  46.. 
lla;4-82/  =  37, 
5a;  +  6  2/=18. 
r2/  +  2a;  =  18, 
[y-2x=.2. 

2  y  ~  3  X  =  5, 
5?/  +  4a;  =  93. 
4:X-7y  =  12, 

3  x  +  5y  =  50. 
8a;  +  72/  =  37, 
4a;-32/=  -1. 
lla;-52/  =  29, 

3aj  +  22/  =  18. 


POSITIVE   AND   NEGATIVE  NUMBERS  77 

Problems 

104.   1.   The  sum  of  two  numbers  is  8  and  their  difference 
is  2.     Find  the  numbers. 


Solution 

Let 

X  =  the  larger  number. 

and 

y  =  the  smaller  number. 

Then, 

x-\-y  =  S, 

(1) 

and 

x-y  =  2. 

(2) 

Adding  (1)  and 

(2), 

2x  =  lO,  .■.x  =  6. 

Subtracting  (2) 

from 

(1),    2y  =  6;  .-.^  =  3. 

Hence, 

the  numbers 

ire  5  and  3. 

Find  two  numbers  related  to  each  other  as  follows : 

2.  Sum  =  14;  difference  =  8. 

3.  Sum  of  2  times  the  first  and  3  times  the  second  =  34 ; 
sum  of  2  times  the  first  and  5  times  the  second  =  50. 

4.  Sum  =  18 ;  sum  of  the  first  and  2  times  the  second  =  20. 

5.  A  cotton  tent  is  worth  ^10  less  than  a  linen  one  of  the 
same  size,  and  3  cotton  ones  cost  $2  more  than  2  linen  ones. 
Find  the  cost  of  each. 

Solution 

Let  X  =  the  number  of  dollars  a  linen  tent  costs, 

and  y  =  the  number  of  dollars  a  cotton  tent  costs. 

Then,  x-y  =  10,  (1) 

and  3y— 2x  =  2.  (2) 

Multiplying  (1)  by  2, 

2x-2y  =  20.  .  (3) 

Adding  (2)  and  (3),  y  =  22.  (4) 

Substituting  (4)  in  (1), 

X  -  22  =  10 ;  .-.  X  =  32. 

Hence,  a  linen  tent  costs  $  32  and  a  cotton  one  1 22. 

6.  A  steam  train  took  10  minutes  longer  to  pass  through  the 
Simplon  tunnel  than  an  electric  train.  What  was  the  time  of 
each,  if  the  steam  train  lacked  8  minutes  of  taking  twice  as 
long  as  the  electric  train  ? 


78  POSITIVE  AND  NEGATIVE  NUMBERS 

7.  During  one  month  tlie  number  of  arrivals  and  departures 
of  vessels  at  the  port  of  Seattle  was  183.  There  were  5  more 
arrivals  than  departures.     Find  the  number  of  each. 

8.  At  one  time  the  United  States  Navy  had  17  coaling  sta- 
tions on  the  Atlantic  and  Pacific  coasts.  If  the  Atlantic  had 
had  1  less,  it  would  have  had  3  times  as  many  as  the  Pacific 
coast.     How  many  coaling  stations  were  there  on  each  coast  ? 

9.  The  length  of  the  Grand  Canal  in  China  is  13  times  the 
approximate  length  of  the  Panama  Canal,  and  the  difference  in 
their  lengths  is  600  miles.     Find  the  length  of  each. 

10.  A  steam  train  is  25  tons  heavier  than  an  electric  train 
that  carries  as  many  passengers.  If  9  such  steam  trains  weigh 
as  much  as  14  of  the  electric  ones,  find  the  weight  of  each  train. 

11.  If  at  Christmas  time  3  dozen  carnations  and  2  dozen 
orchids  cost  $30,  and  2  dozen  carnations  and  3  dozen  orchids 
cost  $  40,  find  the  cost  of  each  per  dozen. 

12.  Small  goldfish  are  worth  $4  per  hundred  less  than  large 
ones.  If  3  hundred  of  the  former  and  2  hundred  of  the  latter 
together  cost  $  18,  find  the  cost  of  each  per  hundred. 

13.  A  stock  car  will  hold  45  more  sheep  than  hogs.  If  the 
number  of  animals  in  16  cars  of  sheep  is  the  same  as  in  25  cars 
of  hogs,  find  the  number  in  a  car  load  of  each. 

14.  One  sugar  factory  employs  2200  men  in  the  factory  and 
fields.  If  the  number  of  field  hands  is  100  more  than  twice 
the  number  of  factory  workers,  find  the  number  of  each. 

15.  The  Eoosevelt  dam  in  Arizona  is  17  feet  lower  than  the 
Croton  dam,  and  twice  the  height  of  the  latter  plus  3  times 
the  height  of  the  former  is  1434  feet.     Find  the  height  of  each. 

16.  The  duty  paid  on  7  pianos  entering  Italy  was  $173.70. 
If  the  duty  paid  on  each  upright  piano  was  $17.37,  or  i- 
that  on  each  grand  piano,  how  many  pianos  of  each  kind 
were  imported? 


POSITIVE   AND   NEGATIVE  NUMBERS  79 

17.  Prospect  Park  is  326|  acres  smaller  than  Central  Park, 
and  twice  the  area  of  the  former  is  1891  acres  more  than  the 
area  of  the  latter.     Find  the  area  of  each. 

18.  A  woman  picked  5  crates  of  Brussels  sprouts,  containing 
in  all  192  quarts.  If  the  crates  hold  32  and  48  quarts  respec- 
tively, how  many  crates  of  each  size  did  she  pick  ? 

19.  One  year  a  jeweler  had  693  broken  watch  springs  brought 
to  him  to  renew.  The  number  broken  in  summer  lacked  39  of 
being  twice  the  number  broken  in  winter.  How  many  were 
broken  in  summer  ?  in  winter  ? 

20.  An  engine  at  Sharon,  Pa.,  weighed  40  tons  less  than 
5  times  as  much  as  two  of  its  castings.  The  weight  of  the 
whole  engine,  minus  twice  the  weight  of  the  castings,  was  314 
tons.    Find  the  weight  of  the  whole  engine  and  of  the  castings. 

21.  In  a  typewriting  contest  in  Paris  a  woman  in  a  given 
tiitie  wrote  500  words  less  than  a  man,  and  twice  the  number 
that  the  man  wrote  is  15,500  less  than  3  times  the  number  that 
the  woman  wrote.     Find  the  number  of  words  written  by  each. 

22.  In  United  States  money,  2  marks,  German  money,  and 
3  francs,  French  money,  are  valued  at  $1,055,  and  1  mark  and 
5  francs  at  $  1.203.  What  is  the  value  in  United  States  money 
of  a  mark  ?  of  a  franc  ? 

23.  The  expense  of  running  a  small  automobile  is  estimated 
at  51  ^  a  week  more  than  the  expense  of  keeping  a  horse  and 
carriage.  The  former  can  be  run  for  3  weeks  for  $  2.22  less 
than  the  latter  can  be  kept  for  4  weeks.  What  is  the  weekly 
expense  of  each  ? 

Suggestion. — Both  equations  should  be  expressed  in  terms  of  cents, 
or  both  in  terms  of  dollars. 

24.  It  cost  42  cents  to  stop  a  certain  train  and  get  it  back 
to  its  former  speed.  Another  train  of  less  speed  cost  35  cents 
to  stop  and  start.  If  in  all  both  trains  made  5  stops,  at  a  cost 
of  $  1.89,  find  the  number  of  stops  made  by  each. 


80  POSITIVE   AND   NEGATIVE   NUMBERS 

DIVISION 

105.  Sign  of  the  quotient. 

Since  division  is  the  inverse  of  multiplication,  the  following 
are  direct  consequences  of  the  law  of  signs  for  multiplication 
given  in  (§  78) : 

+  a  X  H-  6  =  +  «& ;  .  *.  -\- ab  ^ -{- a  = -{- b. 

-\-ax—b  =  —  ab',  .'.  —  ab-r--\-a  =  —  b. 

—  ax-{-b  =  —  ab;  .•.  —ab-. —  a  =  -\-b. 

—  a  X  —  &  =  +  a&;  .*.  +ab-. —  a  =  —  6. 
Hence,  for  division : 

106.  Law  of  signs.  —  The  sign  of  the  quotient  is  +  ^chen  the 
dividend  and  divisor  have  like  signs,  and  —  whe7i  they  have 
unlike  signs. 

EXERCISES 

107.  1.   Divide  each  of  the  following  numbers  by  2:   . 
6,    -6,   10,    -10,   14,    -12,    -18,   22,    -8. 

2.    Divide  each  of  the  foregoing  numbers  by  —  2. 
Perform  the  indicated  divisions : 


3.    7)-l- 

4 

-4). 

4. 
8. 

-3)15 

22-(- 

2). 

5.   -3)- 

-12 

6. 
10. 

-1)9 

7.   4-5-(- 

9.    -l- 

(-1). 

-6h-3. 

.1.   9^(- 

■3). 

12. 

-21- 

3. 

13.  45^(- 

-5). 

14. 

-8-f-2. 

'■!■ 

16. 

28 

-7 

■'■  =f- 

18. 

-20 
-5 

-T?- 

20. 

48 
-6 

-^- 

22. 

72 
8* 

108.  To  divide  when  the  numbers  are  either  positive  or  negative. 

Division,  when  the  numbers  involved  are  positive,  was  treated 
in  §§  30-38.  The  student  is  now  prepared  to  divide  whether 
the  numbers  are  positive  or  negative,  since  the  only  new  point 
involved  is  the  matter  of  signs,  just  discussed. 


POSITIVE  AND  NEGATIVE  NUMBERS  81 


EXBRCISBS 

109.    1.  Divide  -  8  aV  by  2  aV. 

Explanation. — Since    the  signs  of    dividend  and 
PROCESS         divisor  ai-e  unlike,  the  sign  of  the  quotient  is  —  (Law  of 
2aV)-8aV  Signs,  §  106). 

—  4aa^        8-=-2  =  4  (Law  of  Coefficients,  §  33) . 

a*  -f-  a^  =  a*-^  -a}  =  a  (Law  of  Exponents,  §  32). 
x6  ^  a:4  _  a^-4  -^a  (Law  of  Exponents). 
Hence,  the  quotient  is  —  4  ax^. 

Divide : 


2.   30  m^n^  by  5  mn. 

0.   42  n^x*  by  -  6  w«. 

3.    -24a:22^Vby  8x2y. 

7.    -12pVby  12  p^. 

4.   21  aa^?/  by  —  7  ay. 

8.    -  20  7-V  by  -  10  r^.     * 

5.-9  db<^  by  —  3  abc. 

9.   40  mnx^  by  —  8  mnv. 

10.   Divide  4  a^ft  -  6  a*6*  +  4  a6»  by  2  a6 ;  by  -  2  a6. 

PROCESS 

PROCESS 

2a6)4a»6-6a262^4a6» 

^2  ah)     4a»6-6a^62-|-4a^» 

2a2   -3a6    +2  6^ 

-2a^   -h3a6  -26* 

Test  of  Signs.  —  When  the  divisor  is  positive,  the  signs  of  the  quotient 
should  be  like  those  of  the  dividend.  When  the  divisor  is  negative,  the 
signs  of  the  quotient  should  be  unlike  those  of  the  dividend. 

Divide : 

11.  a^y  -2afhj  ay,  by  -  ay. 

12.  9ar^/  +  15a;3/2by  3xy;  by  -3a2/».. 

13.  —x^  —  ^x^-{-  Q?7?  by  —  icz ;  by  a». 

14.  3ar»-6ic»  +  9a;'-12a^by  3ic2.  by  _3a^ 

15.  30r3s3  +  15?V-45rs<  +  75rby  15r;  by  -15r. 

16.  —  ^u  —  t*uv  +  tu^v  —  ^wV  +^w*V  by  tu]  by  —  ^m. 

17.  af +  2af'+^-5ic«+2-a^+«  +  3a^+*by  aj«;  by  -»•. 

18.  a?"  -  a;"-^  +  a;''^  _  jb»-3  +  a;"-<  -  a;""*  by  ar»;  by  —  a^. 

19.  2/"+^  —  2  2/**+2  +  2/"+^  —  3  2/**+*  +  2/"-^'  by  y""^^ ;  by  —  y**+\ 

20.  a{x  -h  2/)''-  a&(a;  +  yf^  o:'h\x  +  y)*  by  -  a ;  by  a{x  +  y)\ 

MILNE'S    1st   TR.   ALO.-— 6 


82  POSITIVE   AND   NEGATIVE   NUMBERS 

21.   Divide  81  +  9  a'  +  a*  by  a^  _  3  ^  +  9 ;  test  the  result. 


PROCESJ 

5 

TEST 

a'-\-9a^-\-Sl 

a2_3a  +  9 

91  H-     7 

a*-3a^-\-9a' 

a24-3a  +  9 

=  13 

3a«  +  81 

3a3-9a2  4-27a 

9a'-27a  +  Sl 
9a2_27a+81 

Note.  —  The  test  is  made  by  substituting  1  for  a  ;  similarly,  the  result 
may  be  tested  by  substituting  any  other  value  for  a,  except  such  as  gives 
for  the  result  0  -r-  0  or  any  number  divided  by  0,  because  we  are  unable 
to  determine  the  numerical  value  of  such  results. 

Divide,  and  test : 

22.  a*  +  16  +  4a2by  2a  +  a2  +  4. 

23.  a^-61x-60hyx^-2x-S. 

24.  a'^-41a-120by  a2+4a-f5. 

25.  25ar^-a^-8ic-2a;2by  5ar^-4a;. 

26.  a»  +  a«  +  cfc*  +  a2  +  3a-l  by  a  +  1. 

27.  4/-92/'-l+62/by3i/  +  2/-l. 

28.   2a*-5a^b-\-6a^b^-4.ab^'\-b'hy  a'-ab  +  b^ 

PROCESS  TEST 

a2_     ab-j-b^        0-1 


2a^_3a6  +  62  =0 


2  a' - 5  a^b  -^6  a'b^ -4.ab'  +  b' 
2a^-2a^b-{-2a'b' 

-  3  a^ft  +  4  a'b^  -  4  ab^ 

-Sa^-^-Sa'b^-Sab^ 

a'b''-     ab'-^b* 

Note.  —  It  will  be  observed  from  the  test  that  0  -?-  1  =  0.    In  general, 
0  -^  a  =  0  ;  that  is,  zero  divided  by  any  number  equals  zero. 

29.  Divide  aa^  —  aV  —  by?-\-  IP-  by  ax  —  b. 

30.  Divide  20  a^y -25  x^ -IS  f  -^27  xf  hj  6y - 5  x. 

31.  Divide  a*  -  4  a^a;  -f-  6  a V  -^aa^-\-x*  by  a^-2ax-{-x'. 


POSITIVE   AND  NEGATIVE   NUMBERS 


83 


32.   Divide  c^  —  8  by  c  +  2. 

PROCESS 

c»-8 

c»4-2c2 


~2&- 

-8 

-2&- 

-4c 

4c 

-8 

4c 

+  8 

c  +  2 

f  -i-d  -L 

(?-2 

-16 

c  -f-^  -h 

c  +  2 

35.  m'  —  n*  by  m  +  n. 

36.  m*  +  n*  by  m  -f  w. 


-16 
Divide,  and  test  results  : 

33.  ar*  +  32  by  a;  +  2. 

34.  a^-?/«by  ar'  +  Z. 

37.  a;'4-2aj«-2a;*H-2a^-lby  a;  +  l. 

38.  2r'  +  32/*  +  5?/3  +  32/2  +  3?/  +  5by.7-fl. 

39.  2n*-4n*-3n3  +  7n2-3n  +  2by  w-2. 

40.  i/*+7y-102/2_2^^15  by /-2y-a 

41.  7a,'8  +  2a;<-27ar^  +  16-8a5by  ar^  +  5a;-4. 

42.  28a^+6a:8^gaj2_ga._2by  2-|-2a;+4««. 

43.  25v2_20i;8+3'y*-M6v-6by  3'y2-8i;4-2. 

44.  4-18a;+30«2_23aj3^.6a^by2a^-5a;  +  2. 

45.  32a^  +  24aj*-25a;-4-16ar^by  6»»-a;-4. 

46.  1  by  1  -f-  a;  to  five  terms  of  the  quotient. 

47.  1  by  1  —  a;  to  five  terms  of  the  quotient.  » 

48.  a3-6a2-hl2a-8-6'by  a-2-6. 

49.  2/^  +  32a:«by  16a;*-hy*-2a!y»-8a5»yH-4a:«2/2. 

50.  2  -  3  n'  +  13  n^  +  23  w*'  -  11  n^  +  6  n«'  by  2  +  3  n«. 

51.  6a2«  +  5a2«-i_i0a2»'-2  +  20a2— 3-16a2"»-*  by  2  a"*  + 
3a'»-i-4a'^-2. 


84  POSITIVE  AND  NEGATIVE   NUMBERS 

Special  Cases  in  Division 
110.    1.   By  actual  division, 
{x^-y^)-^{x-y)=x  +  y. 
{3?-f)^{x-y)=s?  +  xy-{-f. 
{:(^-y'^)^{x-y)=Q(?  +  x^y  +  xf-\-i^. 

Observe  that  the  difference  of  the  same  powers  of  two  num- 
bers is  divisible  by  the  difference  of  the  numbers. 

'  Divisible '  means  '  exactly  divisible.' 

2.  By  actual  division, 
{x'^-y^)^(x-\-y)=x-y. 

(ar'-y')-J-(ic  +  2/)  =  ar^-a;?/  +  2/^  rem.,  -2f.   ^__ 

(a;*  -  y*)^{x  +  y)^x^-ay^y  +  xy^  -  f. 

(x^  —  y^)  H-  (a;  4-  2/)  =  ic*  —  x^y  +  a^y^  —  xy^  +  y\  rem.,  —  2  ?/*. 

Observe  that  the  difference  of  the  same  powers  of  two  num- 
bers is  divisible  by  the  sum  of  the  numbers  only  when  the  powers 
are  even. 

3.  By  actual  division, 

(o^  +  1/2)  -J-  (a;  -  2/)  =  ^  +  y,  rem.,  2  y^. 
(a^  +  /)  -^(x  —  y)=x'^-+-  xy  -f  y^  rem.,  2  f. 
(x* -\- y^)^(x - y)=  a^  +  x^y -{- xy^  +  f,  rem.,  2y*, 
Observe  that  the  sum  of  the  same  powers  of  two  numbers  is 
not  divisible  by  the  difference  of  the  numbers. 

4.  By  actual  division, 

(x^  +  y^^(x  +  y)=x-y,  rem.,  2y^. 

(x^^  -\-y^)^(x  -{.y)  =  x' -  xy  -\-  y\ 

(x*  +  y*)^{x -^ y)  =  a^ - x^y -\- xy^ -  f,  rem.,  2y*. 

(x^-^f)-i-(x  +  y)=x*  —  x^y+a^y^~xf-\-y*. 

Observe  that  the  sum  of  the  same  powers  of  two  numbers  is 
divisible  by  the  sum  of  the  numbers  only  when  the  powers  are 
odd. 

111.  Hence,  the  preceding  conclusions  may  be  summarized 
as  follows,  n  being  a  positive  integer : 


POSITIVE   AND  NEGATIVE   NUMBERS  85 

Principles.  —  1.   a;*  —  y"  is  always  divisible  by  x  —  y. 

2.   of  —  y""  is  divisible  by  x  +  y  only  when  n  is  even, 

1 3.   aj"  -f  y"  is  never  divisible  by  x  —  y. 

|4.   a;**  +  y"  is  divisible  by  x-\-y  only  when  n  is  odd. 

112.  The  following  law  of  signs  may  be  inferred  readily : 
When  x  —  y  is  the  divisor,  the  signs  in  the  quotient  are  plus. 
When  x-^y  is  the  divisor,  the  signs  in  the  quotient  are  alter- 
nately plus  and  minus. 

113.  The  following  law  of  exponents  also  may  be  inferred : 
In   the  quotient  the   exponent  of  x  decreases  and  that  of  y 

increases  by  1  in  each  successive  term, 

EXERCISES 

114.  Find  quotients  by  inspection : 

gs  —  68  'fR^  —  n^  g^  — 8 

a  —  b  m  —  n  a  — 2 

4.  Devise  a  rule  for  dividing  the  difference  of  the  cubes  of 
any  two  numbers  by  the  difference  of  the  numbers. 

Find  quotients  by  inspection : 

5.  t±^.  6    TLjtJ^.  7    t±IL. 

a+b  '    m+n  '     c+3 

8.  Devise  a  rule  for  dividing  the  sum  of  the  cubes  of  any 
two  numbers  by  the  sum  of  the  numbers. 

Find  quotients  by  inspection : 

9.  t^l?^.  12.    r^izl.  15.    ^+«' 


10.  ^^    •  13. f .  16. 

n-f-4  71  —  1 

11.  Vl±^.      ■  14.   ^3i.  17. 


1  + 

a 

ar'- 

32 

X- 

-2 

a*- 

■81 

a  +  3 


86  POSITIVE   AND  NEGATIVE  NUMBERS 

PARENTHESES 

115.  The  student  has  seen  how  parentheses,  (  ),  are  used  to 
group  numbers  that  are  to  be  regarded  as  a  single  number. 
Other  signs  used  in  the  same  way  are  brackets,  [  ] ;  braces, 
\  \ ;  the  vinculum,  ;  and  the  vertical  bar,  \ . 


Thus,  all  of  the  forms,   (a  +  6)c,  [a  +  fe]c,  {a  +  h]c,  a-\-h-c,  and 


a 

+  6 


c,  have  the  same  meaning. 


These  signs  have  the  general  name,  signs  of  aggregation. 

When  numbers  are  included  by  any  of  the  signs  of  aggregation,  they 
are  commonly  said  to  be  in  parenthesis^  in  a  parenthesis^  or  in  parentheses. 

116.   Removal  of  parentheses  preceded  by  -f-  or  — . 

EXERCISES 

1.  Remove  parentheses  and  simplify  3a  +  (6  +  c  —  2a). 

Solution 

The  given  expression  indicates  that  (6  +  c  —  2  a)  is  to  be  added  to  3  a. 
This  may  be  done  by  writing  the  terms  of  (6  +  c  —  2  a)  after  3  a  in  suc- 
cession, each  with  its  proper  sign,  and  uniting  terms. 

.-.  3a+(&-f-c-2a)=3a-f6  +  c-2a  =  a  +  6  +  c. 

2.  Remove  parentheses  and  simplify  4  a  —  (2  a  —  2  6). 

Solution 

The  given  expression  indicates  that  (+2 a  — 2  6)  is  to  be  subtracted 
from  4rt.  Proceeding  as  in  subtraction,  that  is,  changing  the  sign  of 
each  term  of  the  subtrahend  and  adding,  we  have 

4  a  -  (2  a  -  2  5)  =  4  a  -  2  a  +  2  6  =  2  a  +  2  6. 

Principles.  —  1.  A  parenthesis  preceded  by  a  plus  sign  may 
be  removed  from  an  expression  without  changing  the  signs  of  the 
terms  in  parenthesis. 

2.  A  parenthesis  preceded  by  a  minus  sign  may  be  removed 
from  an  expression,  if  the  signs  of  all  the  terms  in  parenthesis  are 
changed. 


POSITIVE   AND  NEGATIVE  NUMBERS  87 

Simplify  by  removing  parentheses : 

3.  a  -h  (6  —  c).  9.  a  —  m  +  (n  —  m). 

4.  x  —  (y  —  z).  10.  a—b  —  {c  —  d). 

5.  x-{-y-^z).  11.  5a-26-(a-26). 

6.  m  — n  — (— a).  12.  a  — (b  —  c-\- a)  — {c- b). 

7.  m-(7i-2a).  13.  2a^  +  3/-(a^  +  a;i/-2/2). 

8.  5x—(2x  +  y).  14.  m4-(3m— w)  — (2w— m)+ri. 

When  an  expression  contains  parentheses  within  parentheses, 
they  may  be  removed  iw  succession,  beginning  with  either  the 
outermost  or  the  innermost,  preferably  the  latter. 

15.  Simplify  6a;-j3a  +  (9  6-2a)-f-4x-10  65. 

Solution 

6  a;  _  {3  a  +  (9  6  -  2  a)  +  4  X  -  10  6} 
Prin.  I,  =6a:-{3a  +  96-2a  +  4x-106} 

Uniting  terms,  =6x  —  {a  —  b  +  ix} 

Prin.  2.  =6x-a  +  6  — 4x 

Uniting  terms,  =  2  x  —  a  +  ft. 

Simplify : 

16.  Aa-\-b  —  \x-{-4:a-\-b  —  2y  —  (x  —  y)\. 


17.  ab  —  ]ab-\-ac  —  a  —  (2a  —  ac)-\-2a  —  2ac\. 

18.  a  +  [y-\5-{-4:a-6y-rS\-(7y-4:a-l)']. 

19.  4  m  —  [p  -H  3  71  -  (m  +  n)  +  3  —  (6  p  —  3  n  —  5  m)]. 

20.  ab  —  \5  -{•  X -{b  +  c  —  ab  -\-  x)\  -{-[x  —  (b  —  c  —  7)]. 

21.  t- X -  \1  - X -  [1  - X-  (1  - x)  -  {x -1)']  - X  +  1\. 

22.  Simplify  a^-f  a(6-a)-6(26-3aV 

Solution 
The  expression  indicates  the  sum  of  a^,  a(h  —  a),  and  —  6(2  6  —  3  a). 
Expanding,  a(6  —  a)=ab^a^  and  —  6(2  6  —  3  a)  =  —  2  6^  +  3  a6. 
Therefore,  writing  the  terms  in  order  with  their  proper  signs, 
o8  +  a(6-a)-6(2  6-;Ja)=a-^  +  ab  -  a^  -  26=^  +  3a6  =  4a6  -  26^. 


88  POSITIVE  AND  NEGATIVE  NUMBERS 

Simplify : 

23.  x^  +  xijj-x),  26.  ct?-.f-{x-y)\ 

24.  c'-dc-d).  27.  c(a-6)-c(a  +  6). 

25.  5-2(a:-3).  28.  a^ -h'' -'6ah{a-h). 

29.  -2{a^y-xy'')-5{xy^-x'y). 

30.  (3a-2)(2a-3)--6(a-2)(a-l). 

31.  (3m-l)(m4-2)-3m(m  +  3)-h2(m4-l). 

32.  {a-by-2(a^-b^-2a(-a-b)-4:b\ 

33.  (a:^  +  2aryH-2/2)(a:2_2a;y-h2/^-(x'  +  i/^(a^  +  2/^. 

34.  f-l2x^-xy{x^y)-f^  +  2(x-y)(a^^xy  +  y^. 

35.  When  a  =  -  2,  6  =  3,  c  =  4,  find  the  value  of 

a'-'(a-c){b-\-c)-\'2b. 

Solution 

a2_(a_c)(6  +  c)  +  26=(-2)2_(-2-4)(3  +  4)+2.3 
=  4-(-6)(7)  +  6 
=  4_(_42)+6 
=  4  +  42  +  6  =  52. 

When  aj  =  3,  1/  ==  —  4,  2  =  0,  m  =  6,  n  =  2,  find  the  value  of: 

36.  m(x  —  y)  +  z\  39.    (x -\- y)  (m  —  n) -[- 3 z. 

37.  2-fm2-(/-l).  40.    (m-\-xy-(n-yy-y\ 

38.  a:^  —  ?/^  —  771^  +  n^.  41.    xyz  —  n  (x  —  my  —  (nxy. 

42.  3  m  (m  —  w)  +  4  n  (y  —  a;)  —  7  (?/  +  2). 

43.  x^y^(m  —  ny(m-i-n)-i-(m-\-ny(m--n). 

44.  Sm{x  —  y  —  ny  —  (y  —  n  —  x)(n  —  x  —  y). 

45.  (a;  +  2/  +  2!)^-  a;^  (y  +  2:  -  a;)  (a?  +  2  -  2/)  -  2;  (a;  + 1/  -  0). 

46.  (2  a;  +  yy  -(0^-2  yy  -  (m  +  ny  (x-^y  +  z).^ 

47.  {m+n+xy—{m-\-n  —  xy  —  {m—n+xy{—m-\-n-\-xy. 

48.  Show  that  {a-b-^-cy^a" -\-W -{-c^ -2ab  +  2ac~2hc, 
when  a  =  1,  6  =  2,  and  c  =  3 ;  when  a  =  4,  6  =  2,  and  c  =  —  1. 


POSITIVE   AND  NEGATIVE   NUMBERS     .  89 

117.  Grouping  terms  by  means  of  parentheses. 

It  follows  from  §  116  that : 

Principles.  —  1.  Any  number  of  terms  of  an  expression  may 
he  inclosed  in  a  parenthesis  preceded  by  a  plus  sign  without  chang- 
ing the  signs  of  the  terms  to  be  inclosed. 

2.  Any  number  of  terms  of  an  expression  may  be  inclosed  in  a 
parenthesis  preceded  by  a  minus  sign,  if  the  signs  of  the  terms 
to  be  inclosed  are  changed. 

EXERCISES 

118.  1.   In  a^  4-  2  a6  +  6^,  group  the  terms  involving  6. 

Solution 

a2  +  2  «6  +  62  =  a2  +  (2  a6 -f  &2). 

2.  Irv  a^  —  x^  —  2  xy  —  y^,  group  as  a  subtrahend  the  terms 
involving  x  and  y. 

Solution 
a2  _  a;2  -  2  xy  -  y^  =  rt*  -  (a;2  +  2  ary  +  y2). 

3.  In  a.i^  -f-  a5  -t-  2  a^  +  2  6,  group  the  terms  involving  x*,  and 
also  the  terms  involving  6,  as  addends. 

4.  In  a''  +  3  a^b  -h  3  a6^  +  6',  group  the  first  and  fourth  terms, 
and  also  the  second  and  third  terms,  as  addends. 

In  each  of  the  following  expressions  group  the  last  three 
terms  as  a  subtrahend  : 

5.  a''-b^-2bc-c\         7.   a'-\-2ab-\-b''-c^  +  2cd-d\ 

6.  a^-b^  +  2bc-c\         8.   a^ -2ab  +  b^ -(?-2cd-d\ 

9.    In   a'^  +  2aZ;4- 6^  — 4  a  — 4  6  +  4,    group   the   first   three 
terms  as  an  addend  and.  the  fourth  and  fifth  as  a  subtrahend. 
Errors  like  the  following  are  common.     Point  them  out. 

10.  a?-'S?-\-x-'l={3?'-l)-ia?-irx). 

11.  a*~2/^-h22/2-22_a^_(y2^.2y2-20' 


90  POSITIVE   AND  NEGATIVE  NUMBERS 

119.  The  use  of  parentheses  in  grouping  numbers  enables 
us  to  extend  the  application  of  certain  cases  in  multiplication. 

Thus,  in  §  94  and  in  §  97,  one  or  both  numbers  may  consist  of  more 
than  one  term. 

EXERCISES 

120.  1.  Expand  (a  +  m  — 7i)(a  — m  +  w). 

Solution 

a-}-m  —  n  =  a+(m  —  n)  and  a  —  m  +  n  =  a  —  (m  —  w). 
.-.  [a  +  m  —  njla  —  m  +  «]  =  [a  +  (w  — n)][a  —  (m  —  w)] 
§  94,  =  a2  -  (m  -  ny 

§  88,  =  rt2  _  (7^2  _  2  m«  +  n2) 

=  a^—m^-j-2  mn  —  nP: 
Expand : 

2.  (r-\-p-q){r-p  +  q).  5.    {3^-^2x+l){o?+2x-l). 

3.  {r-^p  +  q){r-p-q).  6.    (a^+2a;-l)(a^-2«4-l). 

4.  (a;4-6  4-w)(a;-6-n).  7.    (a^+3  a!-2)(a^-3  aj+2). 

8.  [(a+6)+(c  +  d)][(a  +  6)-(c  +  d)]. 

9.  (a+2>+ a'4-2/)(<^  +  ^  — a?  — 2/)- 

10.  (a  +  6  4-m  — n)(a  4-&  — '^  +  w)- 

11.  (x  —  m  +  y—  n)(x  —  m  —  y  -\-  n). 

12.  (a  —  m  —  b  —  n)(a  +  m—b-\-n). 
13.   Expand  (a;  +  ?/ +  l)(aj  +  2/ —  3). 

Solution 

(X  4-  2/  +  l)(a;  +  2/  -  3)  =(^+1/  +  l)Cx+l/  -  3) 
§97,  =(x  +  yy-2{x-^y)-S 

§  86,  =  a;2  +  2  xy  +  y2_  2  X  -  2  2/  -  3. 

Expand: 

14.  (x-y-2)(x-y-S).       17.    (^*- 2  ^2_5)(^4_  2i2_|_  2). 

15.  (x''-^x-l)(x^  +  x-}-S).     18.    (2s4-3r  +  4)(2s-}-3r-3). 

16.  (m~7i  +  2)(7nr-w-4).     19.    (2  a+5b-\-6)(2  a+5  b-S). 


POSITIVE   AND   NEGATIVE  NUMBERS  91 

121.   Collecting  literal  coefficients. 


EXERCISES 


Add: 

1.            ax 

2.            bm 

3.             —ex 

4.         (t-hr)x 

bx 

—  cm 

-dx 

(t-{-2r)x 

(a  +  b)x 

(b  —  c)m 

—  (c  +  d)x 

(2t-h3r)x 

5.  ax 

6.        cy 

7.   —mp 

8.      (a  -h  b)x 

nx 

-dy 

-np 

(2a-hc)x 

Subtract  the  lower  expression  from  the  upper  one  : 

9.  mx  10.  dy-\-az  11.  ax  — by 

nx  ey  —  bz  2x  —  cy 

12.  a^x-\-aby  13.  mx  —  ny  14.  {2r  —  s)y 

b^x  4-  aby  nx  —  my  (r  4-  2  s)y 

15.  Collect  the  coefficients  of  x  and  of  y  in  ax— ay— bx— by. 

Solution.  —  The  total  coeflBcient  of  x  is  (a  —  b);  the  total  coefficient 
of  y  is  (—  a  —  6),  or  —  (a  +  &). 

.'.  ax  —  ay  —bx  —  by  ={a  —  h)x  —  (a  +  6)y. 

Collect  in  alphabetical  order  the  coefficients  of  x  and  of  y  in 
each  of  the  following,  giving  each  parenthesis  the  sign  of  the 
first  coefficient  to  be  inclosed  therein  : 

16.  ax—by  —  bx—cy  +  dx—ey.  20.  x^+ax—y^-{-ay. 

17.  5  ax-\-3  ay  —  2  dx-\-ny —5  x—y.  21.  a?— ay— ax— y^. 

18.  cx—2bx-{-lay-\-3ax—lx—ty.  22.  bx—cy—2ay-\-by. 

19.  bx-\-cy  —  2ax-\-by—cx—dy.  23.  ra;— ay— sa;H-2cy. 

Group  the  same  powers  of  x  in  each  of  the  following : 

24.  av?  +  b^  —  cx-\-e^  —  dx^  —fx. 

25.  a^-\-3a^-{-Sx-ax^-3a3(^  +  bx. 

26.  a^  —  abx  —  a^—ba^  —  ex  —  mnx^ -\- dx. 

27.  ax*  —  ai^  —  ax^ -{- x^  +  a^  —  x —'  aba^  4-  x^. 


92  POSITIVE   AND  NEGATIVE   NUMBERS 

EQUATIONS  AND  PROBLEMS 

122.    1.    Given  6(2  a;  -  3)  -  7(3  a; -f  5)  ==- 72,  to   find   the 
value  of  X. 

Solution 

6(2  X- 3)  _7(3a;  +  5)  =  -72. 
Expanding,  10  a;  -  16  -  21  x  -  35  =  -  72. 

Transposing,  10  x  -  21  x  =  15  +  .36  —  72. 

Uniting  terms,  —  11  x  =  —  22. 

Multiplying  by  -  1,  11  x  =  22. 

.•.x  =  2. 
Verification.  —  Substituting  2  for  x  in  the  given  equation, 
5(4  -  3)  -  7  (6  +  6)  =  -  72. 
5-77  =-72. 
Hence,  2  is  a  true  value  of  x. 

Find  the  value  of  x,  and  verify  the  result,  in : 

2.  2  =  2a;-5-(a;-3).  4.  1  =  5(2  a;-4) +  5  a;  +  6-. 

3.  10 a? -2(a;- 3)  =22.  5.  7(5-3a;)  =  3(3- 4a;) -1. 

6.  4  a;  —  ar^  =  a;(2  —  a;)  4- 2. 

7.  7(2a;-3)=2-3(2a;  +  l). 

8.  3(2-4a;)-(a;-l)  =  -6. 

9.  ^x^-^{y?-^-\-x-2)=^y?. 

10.  5  +  7(a;-5)==15(a;4-2-36). 

11.  2(a;-5)  +  7  =  a;  +  30-9(a;-3). 

12.  (a?-2)(a;-2)  =  (aj-3)(a;-3)  +  7. 

13.  (a;-4)(a;  +  4)  =  (a;-6)(a;4-5)+25. 

14.  a;2_(2a;  +  3)(2a;-3)-f(2a;-3)2=(a;  +  9)(a;-2)-2, 

15.  3(4-a;)2-2(aj  +  3)  =  (2a;-3)2-(a;  +  2)(aJ-2)+l. 

16.  20(2-a;)+3(a;-7)-2[a;  +  9-3S9-8  +  28n=23. 

17.  (2a;-4)2-25a;-6-3a;(4  +  5)5=4(a;-f  2)2  +  71 

18.  3(aj~7)~(a?-9)  +  136  =  a;-2[3a;  +  4-(2a;  +  6-f-9a?)]. 


POSITIVE   AND   .NEGATIVE   x\ UMBERS  93 

Literal  Equations 
123.    1.  Find  the  value  of  x  in  the  equation  bx  —  b^  =  cx  —  c^. 
Solution 


bz-b^  =  cx-c^. 

Transposing, 

bx-cx=  b'^  -  c2. 

Collecting  coefficients  of  x, 

(6-C)X=62_c2. 

Dividing  by  6  —  c, 

62  _  f  2 

=  b-\-C' 
c 

2.  Find  the  value  of  a?  in  the  equation  x  —  a^=2  —  ax. 

Solution 

X  —  a'  =  2  —  ax. 
Transposing,  ax  +  x  =  a'  +  2. 

Collecting  coefficients  of  x,      (a  4-  l)x  =  a'  +  2. 

Dividing  by  a  +  1,  x  =  ^^-±-?  =  a2  -  a  +  1  +  —^  • 

a+1  a+ 1 

Find  the  value  of  a;  in : 

3.  S(x-a-2b)  =  3b.  7.  a^  -  ax-\-5x  =  7  a-10. 

4.  56=3(2a;-6)-46.  8.  2  m»  -  ma;  + '^a;  -  2  7i»  =  0. 

5.  cx-<:^-(JP-\-dx  =  0.  9.  a^-oa;— 2a6  +  6a;+ft'^=0. 

6.  x—l—c=:cx  —  €^  —  c*.  10.  2?i'^-|-5?i+a;=7i^— 72a;— 2. 

11.  3a6-a2-26a;  =  262_aa;. 

12.  a2a;-a»  +  2a2  +  5a;-oa  +  10  =  0. 

13.  aa;-26j;  +  3ca;  =  a6-26-  +  36c. 

14.  ca;-c*-2c'-2c2  =  2c-a;  +  l. 

15.  9a^  +  4:mx  =  -(3ax-16m^). 

16.  a;  +  6n*- 471^  =  1 -3na;  +  2?}-n«. 

17.  Ti^a;  —  3  wi V  +  na;  4- 3  7?i2  +  a;  =  0. 

18.  a;-362-1926V-4ca;-}-16c2a;  =  0. 

19.  r(x  —  sa;  —  1)  -f  7^(x  —  r  +  s^)=—l  —  x  —  r(sx  + 1)  +  r's*. 

20.  a<-c-aa;-6a;4-cx-6*c  =  2a262  +  c(a;-l)-6Vl-f  c). 


94  POSITIVE   AND  NEGATIVE   NUMBERS 

Algebraic  Representation 

124.    1.   Find  the  value  of  x  that  will  make  ^x  equal  to  48. 

2.  Indicate  the  product  when  the  sum  of  x,  y,  and  —  d  is 
multiplied  by  xy. 

3.  If  a  man  earns  a  dollars  per  month,  and  his  expenses 
are  h  dollars  per  month,  how  much  will  he  save  in  a  year  ? 

4.  Indicate  the  sum  of  x  and  z  multiplied  by  m  times  the 
sum  of  X  and  y. 

5.  From  x  subtract  m  times  the  sum  of  the  squares  of 
(a  4- 6)  and  (a  — 6). 

6.  A  number  x  is  equal  to  {y  —  c)  times  (d  -f  c).  Write  the 
equation. 

7.  What  is  the  number  of  square  rods  in  a  rectangular 
field  whose  length  is  (a  +  h)  rods  and  width  (a  —  h)  rods  ? 

8.  At  a  factory  where  N  persons  were  employed,  the  weekly 
pay  roll  was  P  dollars.  Find  the  average  earnings  of  each 
person  per  week. 

9.  How  many  seconds  are  a; days  +c hours  -^-d  minutes? 

10.  Express  in  cents  the  interest  on  y  dollars  for  x  years,  if 
the  interest  for  one  month  is  z  cents  on  one  dollar. 

11.  If  it  takes  h  men  c  days  to  dig  part  of  a  well,  and  d  men 
e  days  to  finish  it,  how  long  will  it  take  one  man  to  dig  the 
well  alone  ? 

12.  Find  an  expression  for  5  per  cent  oix\  y  per  cent  of  z. 

13.  A  train  ran  M  miles  in  ^  hours  and  m  miles  in  the  suc- 
ceeding h  hours.  Find  its  average  rate  per  hour  during  each 
period  and  during  the  whole  time. 

14.  A  farmer  has  hay  enough  to  last  m  cows  for  n  days. 
How  long  will  it  last  (a  —  6)  cows  ? 

15.  A  dealer  bought  n  50-gallon  barrels  of  paint  at  c  cents 
per  gallon.  He  sold  the  paint  and  gained  g  dollars.  Find  the 
selling  price  per  gallon. 


POSITIVE   AND   NEGATIVE   NUMBERS  95 

Problems 
125.    Solve  the  following  problems  and  verify  the  solutions : 

1.  I  bought  40  stamps  for  95  cents.  If  part  of  them  were 
2-cent  stamps  and  part  3-cent  stamps,  how  many  of  each  did  I 
buy  ? 

Solution 

Let  X  =  the  number  of  2-cent  stamps. 

Then,  40  —  x  =  the  number  of  3-cent  stamps. 

.-.  2a;  +  3(40-a;)=95. 

Solving,  X  =  25,  the  number  of  2-cent  stamps, 

and  40  —  x  =  15,  the  number  of  3-cent  stamps. 

Verification.  —  The  results  obtained  may  satisfy  the  equation  of  the 
problem  and  still  be  incorrect,  because  the  equation  may  be  incorrect. 
If,  however,  the  results  satisfy  the  conditions  of  the  problem^  the  solution 
is  presumably  con-ect. 

1st  condition  :  The  whole  number  of  stamps  bought  is  40. 
26  +  16  =  40. 

2d  condition  :  The  total  cost  of  the  stamps  =  95  ^. 
The  cost  of  25  stamps  @  2  ^  -|-  15  stamps  @  3  ^  =  95  ^. 

2.  A  certain  paper  mill  produces  350  tons  of  paper  from 
sawdust  each  week.  Of  this  50  tons  more  is  used  for  news- 
papers than  for  wrapping  paper.  How  many  tons  are  used  for 
each? 

3.  The  roadway  of  the  Connecticut  Avenue  concrete  bridge 
in  Washington,  D.C.,  together  with  two  sidewalks,  is  52  feet 
wide.  How  wide  is  the  roadway,  if  it  is  8  feet  less  than  twice 
the  combined  width  of  the  sidewalks  ? 

4.  One  year  the  box  factories  of  New  England  used  6,000,000 
feet  of  boards.  The  amount  of  white  pine  used  less  1,200,000 
feet  was  3  times  that  of  the  other  timber.  How  much  white 
pine  was  used  ? 

5.  It  costs  2i^  more  a  day  to  feed  an  immigrant  than  it 
does  to  feed  a  United  States  private  soldier.  If  it  costs  as 
much  to  feed  44  immigrants  as  it  does  to  feed  49  privates,  find 
the  cost  of  the  daily  rations  of  each. 


96  POSITIVE   AND  NEGATIVE   NUMBERS 

6.  Of  the  160,000  inhabitants  of  Hawaii,  twice  as  many 
were  Japanese  as  Chinese.  The  rest  of  the  inhabitants,  or  J 
of  the  total,  were  Americans  and  Europeans.  Find  the  num- 
ber of  Chinese. 

7.  The  combined  capacity  of  two  ice  factories  is  264  tons  a 
day.  If  the  capacity  of  the  smaller  one  is  increased  57  tons, 
its  capacity  will  be  half  that  of  the  larger  one.  Find  the 
capacity  of  each. 

8.  It  cost  a  man  60^  to  send  a  telegram  at  *30-2',  that  is, 
30^  for  the  first  10  words  and  2^  for  each  additional  word. 
How  many  words  did  the  message  contain  ? 

Suggestion.  —  Let  a;  be  the  number  of  words  in  the  message. 
Then,  a;  —  10  -will  represent  the  number  of  words  in  excess  of  10  words. 
.-.  30  +  2(x~10)  =60 

9.  How  many  words  can  be  sent  by  telegraph  from  New 
Haven  to  New  York  for  75  t  at  the  day  rate,  '  25-2 '  ? 

10.  A  long-distance  telephone  message  cost  me  $1.25.  The 
rate  was  50  f  for  the  first  3  minutes  and  15  ^  for  each  additional 
minute.     How  long  did  the  conversation  last  ? 

11.  The  day  rate  for  a  telegram  between  New  Orleans  and 
New  York  is  '60-4'  and  the  night  rate  is  '40-3.'  A  message 
of  a  certain  number  of  words  cost  25^  less  to  send,  at  night 
than  in  the  daytime.     Find  the  number  of  words. 

12.  A  boy  was  twice  as  old  as  his  sister  4  years  ago.  Now 
his  sister  is  |  as  old  as  he  is.     Find  the  age  of  each. 

13.  During  one  month  the  Dead  Letter  Office  received 
1,000,000  pieces  of  mail  matter.  If  the  number  remaining  in 
the  office  was  \  as  many  as  the  number  returned  to  the  senders, 
how  many  pieces  were  returned  ? 

14.  An  eighteen-hour  train  between  New  York  and  Chicago 
was  late  91  times  during  its  first  year's  run.  It  Avas  late  at 
Chicago  10  times  more  than  50  %  as  many  times  as  it  was  late 
at  Jersey  City.     How  many  times  was  it  late  at  Jersey  City  ? 


POSITIVE   AND  NEGATIVE   NUMBERS  97 

15.  In  China,  one  \^oinan  earned  3^  and  another  8^  a  day 
by  embroidering.  The  former  worked  28  days  on  a  piece  of 
work,  and  then  the  two  finished  it.  If  the  labor  cost  ^5.02, 
how  long  did  each  work  ? 

16.  The  shed  that  sheltered  an  airship  was  544  feet  in 
perimeter.  If  twice  its  length  was  52  feet  more  than  4  times 
its  width,  what  was  its  width  ?  its  length  ? 

17.  The  average  life  of  5-dollar  bills  is  |  of  a  year  longer  than 
that  of  1-dollar  bills,  and  |  as  long  as  that  of  lO-dollar  bills.  If 
a  10-dollar  bill  lasts  If  years  longer  than  a  1-dollar  bill,  find 
the  average  life  of  a  bill  of  each  denomination. 

18.  A  farmer's  net  receipts  from  hens  in  a  year  were  $  90.15. 
The  eggs  sold  for  $92.55  more  than  the  chickens,  and  the  ex- 
penses were  $  72.65  less  than  the  selling  price  of  the  eggs. 
What  did  the  eggs  sell  for  ?  the  chickens  ? 

19.  Upon  the  floor  of  a  room  4  feet  longer  than  it  is  wide  is 
laid  a  rug  whose  area  is  112  square  feet  less  than  the  area  of 
the  floor.  There  are  2  feet  of  bare  floor  on  each  side  of  the 
rug.     What  is  the  area  of  the  floor  ?  of  the  rug  ? 

20.  A  party  of  8  traveled  second  class  from  London  to 
Paris  for  $5.70  less  than  twice  the  amount  paid  by  a  party 
of  3  traveling  first  class.  If  a  first-class  ticket  cost  $4.15 
more  than  a  second-class  ticket,  find  the  price  of  each. 

21.  A  military  cable  and  telegraph  system  between  Seattle 
and  Alaska  covers  4044  miles.  The  length  of  the  submarine 
cable  is  272  miles  less  than  twice  that  of  the  land  telegraph. 
The  land  telegraph  is  12  miles  longer  than  13  times  the  wire- 
less.    How  long  is  the  wireless  ? 

22.  The  United  States  has  280  life-saving  stations,  1  being 
situated  at  the  falls  of  the  Ohio  River.  Of  the  remainder,  the 
Atlantic  coast  has  11 J  times  as  many  as  the  Pacific.  Find  the 
number  on  the  Pacific  coast,  if  it  lacks  2  of  being  ^  the  num- 
ber on  the  Great  Lakes. 

MILNU^S    1st    YR.    ALQ. — 7 


98  REVIEW 


REVIEW 


126.    1.   What  are   positive   numbers?   negative   numbers? 

In  the  following  expression  point  out  the  positive  numbers ; 
the  negative  numbers.     Perform  the  indicated  operations  : 
3ax-]-7by —  9  bx-\- 10  by  —  4:  ax  — 3  bx-\- A  ax  — 2  ax— 12  by. 

2.  What  two  meanings  has  the  minus  sign  in  algebra? 
If  distance  north  is  positive,  what  is  the  meaning  of  — 150  miles  ? 
+  75  miles  ? 

3.  Distinguish  between  arithmetical  numbers  and  algebraic 
numbers. 

4.  Instead  of  subtracting  a  number  (positive  or  negative), 
what  may  be  done  to  secure  the  same  result  ?  Illustrate  by- 
subtracting  —  7  from  + 12.  What  is  the  absolute  value  of 
each  of  these  numbers  ? 

5.  What  is  transposition  ?  Give  the  principle  relating  to 
transposition. 

6.  State  the  law  of  signs  for  multiplication  ;  for  division. 

.  7.  What  is  the  sign  of  the  product  of  an  even  number  of 
negative  factors  ?  of  an  odd  number  of  negative  factors  ? 

8.  In  what  respect  do  (a  —  b)  and  (b  —  a)  differ  ?  Expand 
(a  —  by  and  (b  —  af  and  compare  the  results. 

9.  For  what  values  of  n  is  ic"  +  ?/"  divisible  hj  x  +y?  by 
x  —  y?     When  is  x""  —  ?/"  divisible  hj  x  +  y?  hy  x  —y? 

10.  State  the  law  of  signs  for  the  quotient  when  x""  -f-  y""  or 
of  —  2/"  is  divided  by  x-^y  or  x  —  y  -,  the  law  of  exponents. 

11.  What  must  be  added  to  cc^  — 10  x  to  make  it  the  square 
of  x  —  5?  to  a^  +  b^  to  make  it  the  square  of  a  -f  6  ?  to 
X*  4-  icy  +  2/*  to  make  it  the  square  of  x^  -f  y^  ? 

12.  How  may  a  parenthesis  preceded  by  a  minus  sign  be 
removed  from  an  algebraic  expression  without  changing  the 
value  of  the  expression  ? 


REVIEW  99 

13.  Add  3  a +  56 -lie,  6-2a+c,  2c-\-8a—b,  7c—b+6a, 
5  6  —  4  a  —  2  c,  6  —  a,  c-\-b  ~a,  and  c  —  4  a. 

14.  Subtract  the  sum  of  x  —  2y-}-Sz  —  5w  and  7  a;  -|-  i«  —  2  2 
from  10  X  —  y-\-z  —  Sw. 

15.  If  a;  =  7-^  +  rs  —  S-,  y=2r^+4rs-|-2s^,  smd  z=r^—Srs—^j 
find  the  value  of  a;  +  2/  ~  ^• 

Expand,  and  test  each  result : 

16.  (r^-hTr's-Sr^-\-2s^(r^-\-2rs-\-^. 

17.  (3P  +  6Z2'-m-12Z'-m2-f 3m3)(4^  +  3^m  +  2m2). 

18.  (x*4-y'-4:xf-{-5a^y'-hSa^y)(a^-\-Sa^y  +  3xy'-hf). 
Expand  by  inspection,  and  test  each  result: 

19.  (Sa-hTby.  23.    (7  r +  4s)(7r-4s). 

20.  (9w-2vy,  24.    (Sx-5y  +  zy. 

21.  (x-{-2y)(x-2y).  25.    (2c +d)(3c  +  2d). 

22.  (a-3)(a  +  10).  26.    {oa-3b)(2a  +  2b). 

Divide,  and  test  each  result : 

27.  2Z«-f-5ZV-3^V-6Z»r3H-3Zr^ -?•«  by  2^-36- -h?-2. 

28.  3x^-{-Sy^-10yz-Sa^z-Sz^-\-10a^y  by  ar-\-2y-3z. 

29.  4  aj^"  -  25  aj^Y"  - 10  a^V*  -  2/*"  by  2  ar^"  -  5  xy  -  fy 
Find  quotients  by  inspection : 


30. 

2/  +  1 

32.    ^'^^^. 
a  +  2 

3^    c3  +  125fr^ 
c  +  5(i 

31. 

.^3-64 
a;-4 

33.  ^^-^^y\ 

3x  +  4:y 

__    243 -a;» 
35. 

3  — a; 

36. 

Simplify  17  a;- 

-i3y  +  4.z-[z  +  5 

a  + 

a;-3a-22/]j. 

37.    Simplify  «  +  26-[4c  +  2(a  +  2&)-6  +  4c-a]  +  6. 


100 


REVIEW 


When  a;  =  2, 2/  =  —  3,  and  2  =  5,  find  the  value  of : 

38.  xz  —  (x  +  y-{-z).  40.    x^ —  3x{y  +  z)-\-y^— z. 

39.  3(x-y)+2(y-x)-zy.      41.    (x-y)(y +  z)-z\y -z). 

42.  When  a  =  2  and  6  =  3,  prove  that 

b(ab  +  b  -2  d)=  ab''  -{-b'  -2  ab. 

43.  Collect  similar  terms  within  parentheses  : 

aoc^  —  cy-^CLX  —  2  ax^  -\-2cy^  —  ax  —  cy^  -f  ax^  +  cy. 

44.  Collect  the  coefficients  of  x  and  of  y  in 

7  ax  —  8  by  —  22  a^x  -^  o  ay  — 17  bx  -^  cy  —  4tx  -^  13  y. 
Solve  for  x,  and  test  results : 

45.  3«H-7(a;-2)-13  =  12-3a;. 

46.  20  =  7-5(3-a;)  +  9(a;  +  2). 

47.  x^  —  1  =x{oi^  —  x)  -{-  X  +  3  -{-  x^. 

48.  (x-A)(x-\-3)  =  (x-{-6y-3  +  2x. 

49.^  Aa  —  3{b  +  x)—  5 a  =  7  b  +  4:a  —  5(x  +  a). 

50.    (a  —  x) (b  —  x)  —  b\x  —  (a  —  x)  —  xl  =  a^  —  2 bx -{- ab. 

Solve,  and  test : 


51. 


'3x-{-2y  =  13y 
2x  +  y  =  S. 


gg^    (7x  +  Ay  =  39, 
I        a;-?/ =  4. 


-2/  =  3, 
2.7=13. 


53. 


I3aj- 


54. 


55. 


56. 


'  5x-\-7y  =  24:, 
3x-2y  =  2. 

3x+5y  =  13, 
[2x-3y  =  -4.. 

(9x-2y  =  W, 

[5x-\-7y  =  57. 


Supply  the  missing  coefficients  in  the  following  equations 

57.  3a-*bi-6a  +  Bb-*xy  =  *a-\-b-2xy. 

58.  x'-\-2xy-{-3f-[2a^-h*f-]=*o'^-\-*xy. 
69.   6m2-f9mri-3n2-[3m2+*mn]+w2 


FACTORING 


127.  In  multiplicatwriy  we  find  the  product  of  two  or  more 
given  numbers ;  in  factoring,  we  have  given  the  product  to  find 
the  numbers  that  were  multiplied  to  produce  it. 

These  numbers  are  called  the  factors  of  the  product. 

128.  A  number  that  has  no  factors  except  itself  and  1  is 
called  a  prime  number. 

MONOMIALS 

129.  To  factor  a  monomial. 

While  in  factoring  it  is  usually  the  prime  factors  that  are 
sought,  this  is  not  ge;ierally  true  in  the  case  of  monomials,  be- 
cause the  factors  of  a  monomial,  except  those  of  the  coefficient, 
are  evident. 

Thus,  2rt852  shows  its  prime  factors  as  well  as  though  written 
2  .  a  •  a  •  a  •  6  •  6,  but  81  a^h"^  should  be  written  3*  a^h'^  to  be  considered 
in  factored  form. 

However,  it  is  often  desirable  to  separate  a  monomial  into 
two  factors,  one  being  given  or  both  being  specified  in  some 
way. 

EXERCISES 

130.  1.  In  each  of  the  following,  if  ocy  is  one  factor,  find  the 
other :  6  sc^y,  15  x^y"',  2  ory,  a^x-b-y^,  —  mnxy,  —  xy. 

2.  In  each  of  the  following,  if  ahc  is  one  factor,  find  the 
other:  c^hc,  ah\  ahc\  -  a-tV,  -  a'hc,  -\ahc. 

3.  Find  two  equal  positive  factors  of  a;';  of  9aV;  of  64?7i*. 

4.  Find  two  equal  negative  factors  of  25  a^ ;  of  16  a^ ;  of  9  a^ 

101 


102  FACTORING 

131.  A  factor  of  two  or  more  numbers  is  called  a  common 
factor  of  them. 

132.  One  of  the  two  equal  factors  of  a  number  is  called  its 
square  root. 

Every  number  has  two  square  roots,  one  positive  and  the 
other  negative. 

The  square  root  of  25  is  5  or  —  5,'for  5  •  5  =  25  and  (— 5)(—  5)  =  25. 

In  factoring,  usually  only  the  positive  square  root  is  taken. 

133.  To  factor  an  expression  whose  terms  have  a   common 
monomial  fagtor. 

EXERCISES 

1.   Find  the  factors  ot  Sxy  —  6x^y +  9  xy^. 

Explanation.  —  Bv   examining    the 

PROfPSS 

terms  of>Jihe  expression,  it  is  seen  that 
S  xy  —  6  x"^!/ -\- 9  xy^  the  monomial  Sxy  is  a,  factor  of  every 

=  Sxy(l  —2x-\-  3y)  term.     Dividing  by  this  common  factor 

gives  the  other  factor. 
Hence,  the  factors  of  Sxy  —  6 z^y  +  9 xy'^  are  3 xy  and  1  —  2x  +  Zy. 
Test.  —  The  product  of  the  factors  should  equal  the  given  expression ; 
thus,  Sxy(l-2x  +  3y)  =  Sxy  -  6x^y  +  9xy^. 

Factor,  and  test  each  result : 

2.  5af-5a^.  12.  a^2  _^  a^i  +  a^i^  _  a^. 

3.  Sx^  -\-2x\  13.  ac  —  be  —  cy  —  abc. 

4.  Sx^-6x\  14.  3a^f-3x^y'  +  12xy. 

5.  4a2-6a6.  15.  3m^- 12mV  +  6  mn^ 

6.  5m2-3mn.  16.  9  a%  -  18  a^V  +  24  a^^y. 

7.  Sa^y^-Sa^f.  17.  12  x'yh^  -  W  x^yh' -  20  x^y^^. 

8.  4:a%-6a^b\  18.  25  c^da^  +  35  c«d V  -  55  c^c? V. 

9.  5m*7i-10mV.  19.  16  a^dV  -  24  a^W  +  32  a^^V. 
10.   3a^--9a^-6a;^.  20.  14  a^mw^  -  21  a^mV  -  49  a^wl 
11.3  a>  -  2  a%  +  aK  21.  60  m*nV  -  45  m^^iV  -f-  90  mM^r^. 


FACTORING  103 

BINOMIALS 

134.  To  factor  the  difference  of  two  squares. 

By  multiplication,  §  94,  (a  +  6)  (a  -  6)  =  a^  -  h\ 

Therefore,  a^  -  6^  =  (a  +  b)  (a  -  b). 

Hence,  to  factor  the  difference  of  two  squares, 

Rule. — Find  the  square  roots  of  the  tico  terms,  and  make 
their  sum  one  factor  and  their  difference  the  other. 

EXERCISES 

135.  Factor,  and  test  each  result : 

1.  x'-m^  3.   a^-1.  5.    n^-4?. 

2.  a^-y\  4.    a^-^\  6.   1--- 

7.  Factor  aV- 4  c^. 

Solution 

a2x2-4c2  =  (aa;)2_(2c)2 

=  (ax  +  2  c)  (ax -2  c). 

Factor,  and  test  each  result : 

8.  a2_81.  12.    a^-h\  16.    l-lUm\ 

9.  6* -49.  13.   m'-n^.  17.    Q^a^-a'c^. 

10.  25  2/2-1.  14.    81-icy-  18-    81  a^- 100. 

11.  m^-16ri2.  15.   9  a^- 49  61  19.    121  ?r- 367^. 

136.  To  factor  the  sum  or  the  difference  of  two  cubes. 
By  applying  the  principles  of  §§  111-113, 

^f!±^  =  a^  _  a6  +  6^  and  ^^^^=^' =  a^  +  a.6  4- 6'. 
a-\-b  a  —  b 

Then,  §  30,      a'  +  b'  =  (a-\-b)  (a'  -ab  +  b% 

and  a^-b^  =  {a  -  b)(a^-\-ab  +  b"). 

By  use  of  these  forms  any  expression  that  can  be  written  as 
the  sura  or  the  difference  of  two  cubes  may  be  factored. 


104  FACTORING 

EXERCISES 

137.  Factor,  and  test  each  result : 

1.  a^  +  iyl  3.    m^-l.  5.    7^  +  2^ 

2.  a?-f.  4.    l  +  m\  6.    --1. 

8 

7.  Factor  x^  +  /. 

Solution 

««  +  ys  =  {x^y  +  (2/"^)3  =  (x2  +  y^)  (x*  -  xV  + 1)' 

8.  Factor  a»- 8  61 

Solution 
a9 -8&^=(a3)3 -(26)3  =:(a3-26)(a6  +  2a3&  + 462). 

Factor,  and  test  each  result : 

9.  o?-f.  11.   a^63-27.  13.    aj^/g^+l. 
10.   8  7^  +  s^                12.    v^-\-(jU\               14     w^-lOOO. 

By  applying  §§  111-113,  as  in  §  136,  any  expression  that 
can  be  written  as  the  sum  or  the  difference  of  the  same  odd 
powers  of  two  numbers  may  be  resolved  into  two  factors. 

Thus,  a^  +  6-5  =  (a  +  6)  (a*  -  a'^b  +  a^-h'-  -  ab^  -\-  6*), 

and  a5  -  65  =  (a  -  6)  (a*  +  a^b  +  a^-b^  +  ab^  +  64). 

Factor : 

15.  m^  4-  n^  17.    a^  +  32.  19.    1  -f  s*^. 

16.  m^-7i*.  18.    32 -a*.  20.   x^-y^^. 

TRINOMIALS 

138.  To  factor  a  trinomial  that  is  a  perfect  square. 

By  multiplication,  §  85, 

(a-^b)(a  +  b)  =  a'-\-2ab-hb^ 
Then,  a^^2ab  +  b'  =  (a  +  b)(a  +  by 

Also,  §  88,        (a  -6) (a  -  6)  =  a2  _  2 a6  +  h\ 
Then,  a^  -  2  a6  +  62  =  (a  -  6)  (a  "  6). 


FACTORING  106 

These  trinomials,  o?  +  2  ab  4-  b^  and  a^  —  2  a6  +  ^S  are  perfect 
squares,  for  each  may  be  separated  into  two  equal  factors. 

They  are  types,  showing  the  form  of  all  trinomial  squares, 
for  a  and  b  may  represent  any  two  numbers. 

Hence,  to  factor  a  trinomial  square, 

Rule.  —  Connect  the  square  roots  of  the  terms  that  are  squares 
with  the  sign  of  the  other  term,  and  indicate  that  the  result  is  to 
be  taken  twice  as  a  factor. 


139.  Factor,  and  test  each  result : 

1.  x'-\-2xy-}-y\  3.    (^ -h  2  cd -h  d'.  5.    ar'-2a;4-l. 

2.  p^  —  2pq-\-  q\  4.    t^  —2  tu  +  u^  6.    x^  -\-2x  +  l. 

140.  From  the  forms  in  the  preceding  discussion  and  exer- 
cises it  is  seen  that  a  trinomial  is  a  perfect  square,  if  these 
two  conditions  are  fulfilled : 

1.  Two  terms,  as  -\-a^  and  -\-b^,  must  be  perfect  squares. 

2.  The  other  term  must  be  numerically  equal  to  twice  the 
product  of  the  square  roots  of  the  terms  that  are  squares. 

Thus,  25  a-2  -  20  xy  +  4  y"^  is  a  perfect  square,  for  26x2  =  (6  ic)2, 4  y* 
=  (2  «/)2,  and  -  20  xy  =  -  2(5  «) (2 y). 

EXERCISES 

141.  Discover  which  of  the  following  are  perfect  squares, 
factor  such  as  are,  and  test  each  result : 

1.  a^  +  6x-\-9.  8.  Sx^-\-Sxy-{-2y\ 

2.  4-4a  +  al  9.  16p2_24p  +  9. 

3.  7-2- 8  r  4- 16.  10.  x*-\-2xY-h?A 

4.  m^-mn-\-n\  11.  9  +  42 />»  +  49  &«. 

5.  1+46  +  461  12.  l-6m*  +  9m«. 

6.  l-ea'-f-Oa^.  13.  4:  x^y^  -  20  xy -{- 25. 

7.  m*-f  m-fj.  14.  4  .t^ -f  12  a;]/« -h  9  .?/V. 


106  ^  FACTORING 

142.  To  factor  a  trinomial  of  the  form  jr^  +/?jr  -f  q. 
By  multiplication,  §  97, 

Then,  Jr^  +  (a  4-  b)x  +  ab=  {x  -\-  a)(x  +  6). 

This  trinomial  consists  of  an  a;^-term,  an  oj-term,  and  a  term 
without  X,  that  is,  an  absolute  term.  Therefore,  it  has  the  type 
form  x^  -\-  px  -\-  q. 

Hence,  if  a  trinomial  of  this  form  is  factorable,  it  may  be 
factored  by  finding  two  factoids  of  q  (the  absolute  term)  such  that 
their  sum  is  p  (the  coefficient  of  x),  and  adding  each  factor  of  q  to  x. 

Thus,  x--\-Sx  +  15  =  (x-}-8)(x -\-5), 

a:^-Sx-{-15  =  (x-8)(x-5), 
x'  +  2x-lo^(x~S)(x-{-5), 
r^-2x-lo  =  (x-\-3){x-5). 

EXERCISES 

143.  1.   Resolve  ar'  — 13  a;  —  48  into  two  binomial  factors. 

Solution.  —  The  first  term  of  each  factor  is  evidently  x. 

Since  the  product  of  the  second  terms  of  the  two  binomial  factors  is 
—  48,  the  second  terms  must  have  opposite  signs ;  and  since  their  alge- 
braic sura,  —  13,  is  negative,  the  negative  term  must  be  numerically- 
larger  than  the  positive  term. 

The  two  factors  of  —  48  whose  sum  is  negative  may  be  1  and  —  48, 
2  and  —  24,  3  and  —  16,  4  and  —  12,  or  6  and  —  8.     Since  the  algebraic 
sum  of  3  and  —  16  is  —  13,  3  and  —  16  are  the  factors  of  —  48  sought. 
...  a;2_13x-48=(x  +  3)(ic-16). 

2.   Factor  72  —  m^  —  m. 

Solution.  —  Arranging  the  trinomial  according  to   the    descending 

powers  of  wi, 

72  -  m2  -  wi  =  -  m2  -  m  +  72 

Making  m^  positive,  =  -  (m^  -f-  »w  —  72) 

=  -(m-8)(m +  9) 
=  ( -  wi  +  8)  (m  +  ft) 
=  (8 -m)(9-f- m). 


FACTORING  107 

Separate   into   factors,  and   test    each    result   by   assigning 

a  numerical  value  to  each  letter: 

3.  a^-f7a;  +  12.  13.  x^ -{- 5  ax -{- 6  a^. 

4.  2/2-72/H-12.  14.  x^-6ax-\-5a\ 

5.  p'-S2J  +  12.  15.  y' -4.by- 12  b\ 

6.  r2  +  8r  +  12.  *                16.  f-3ny-2Sn\ 

7.  15-\-2a-a\  17.  z'-anz-2a^nK 

8.  b^-\-b-12.  18.  -a^+25a;-100. 

9.  30-?*^  +  r.  19.  X* -{- 19  ca:^-\- 90  c^. 

10.  c^-c- 72.  20.   a.-^ 4- 12 aa^  + 20  a^. 

11.  c2-5c-14.  21.   a;i«-116V  +  24&*. 

12.  aj2-a;-110.  22.    n^a:^  _  i;|^  ^^^  _,_  3q  yj 

144.    To  factor  a  trinomial  of  the  form  ax^  4-  6x  -f-  c. 

EXERCISES 

1.   Factor  3  a^  4- 11  a;  -  4. 

Solution.  — If  this  trinomial  is  the  product  of  two  binomial  factors, 
they  may  be  found  by  reversing  the  process  of  multiplication  illustrated 
in  exercise  32,  page  73. 

Since  3  x^  is  the  product  of  the  Jirst  terms  of  the  binomial  factors,  the 
first  terras,  each  containing  x,  are  Sx  and  x. 

Since  —  4  is  the  product  of  the  last  terms,  §  78,  they  must  have  unlike 
signs,  and  the  only  possible  last  terms  are  4  and  —1,-4  and  1.  or  2 
and  -  2. 

Hence,  associating  tliese  pairs  of  factors  of  —  4  with  3  x  and  x  in  all 
possible  ways,  the  possible  binomial  factors  of  3  sc^  -|-  11  a:  —  4  are  : 

3  a; +  4  1        3.r-ll        3x-4  1        3a;+11       3a--f2|        3x-21 
jB  -  1  j  '        a!  +  4  J  '       a-  +  1  I  a;  -  4  J  .r  -  2  j  '        a;  +  2  J 

Of  these  we  select  by  trial  the  pair  that  will  give  + 11  a;  (the  middle 
term  of  the  given  trinomial)  for  the  algebraic  sum  of  the  *  cross-products,' 
that  is,  the  second  pair. 

.-.  3 a;2  +  11  X  -  4  =  (3  ar  -  l)(x  4-  4). 


108  FACTORING 

By  a  reversal  of  the  law  of  signs  for  multiplication  and  from 
the  preceding  solution  it  may  be  observed  that : 

1.  When  the  sign  of  the  last  term  of  the  trinomial  is  4-,  the  last 
terms  of  the  factors  must  be  both  -f-  or  both  — ,  and  like  the  sign 
of  the  middle  term  of  the  trinomial. 

2.  When  the  sign  of  the  last  term  of  the  trinomial  is  — ,  the 
sign  of  the  last  term  of  one  factor  must  be  +,  and  of  the  other  — . 

Factor,  and  test  each  result : 

2.  2?/2  +  3?/  +  l.  10.  2x^-{-x-15. 

3.  5x^-\-9x-2.  11.  5a^-\-lSx-\-6. 

4.  Sx'-Tx-G.  12.  3a^-17a;  +  10. 
6.   4r2_^3y._|_3  13  ex^-nx-S5. 

6.  6x^-7x-{-2.  14.  15x;^  +  17x-4:. 

7.  2a^-5x-12.  15.  Wx'-Ux-S. 

8.  lOt^'  +  t-S.  16.  2x'-\-3xy-2y\ 

9.  en^-lSn  +  Cy.  17.  3x^-10xy -\-3y\ 

When  the  coefficient  of  a:^  is  a  square,  and  when  the  square 
root  of  the  coefficient  of  x^  is  exactly  contained  in  the  coeffi- 
cient of  X,  the  trinomial  may  be  factored  as  follows : 

18.  Factor  9  a;2  + 30  a; +  16. 

Solution 

=  (3a:)2+10(.3a:)+16 
=  (3a;  +  2)(3x  +  8). 

Separate  into  factors,  and  test  each  result : 

19.  9a^-9a;  +  2.  25.  4.9x^-i2x-55. . 

20.  4.2(^-4.x-W.  26.  25a^4-25x-24 

21.  9a^-42a;  +  40.  27.  l(ja^-S2x-hl5. 

22.  25  a^ +  15  a; -f- 2.  28.  64 ar^-~ 32  a? -77. 

23.  16a:2  +  16a;  +  3.  29.  100a^  +  40x  +  3. 

24.  sear' -36 a: +  5.  30.  81  a:^- 108 a: +35. 


FACTORING  109 

POLYNOMIALS 

145.  To  factor  a  polynomial  whose  terms  may  be  grouped  to 
show  a  common  polynomial  factor. 

EXERCISES 

1.  Factor  ax -\- ay -{- bx -{-  by. 

Solution 

ax  +  ay-h  bx  +  by  =  (ax  +  ay)  +  (bx  +  by) 
=  a(x -^  y) -{- b(x -\- y) 
=  (a  +  6)(x  +  i/). 

2.  Factor  ax  -\-  by  —  ay  —  bx. 

Solution 

ax  +  by  —  ay  —  bx=:ax—  ay  —  bx  +  by 
§  117,  Prin.  2,  =  (aa;  -  ay)  -  {bx  -  by) 

=  a{x-y)-h{x-y) 
=  (a-b)(x-y). 

Remark. — The  given  polynomial  must  be  arranged  and  grouped  in 
such  a  way  that  after  the  monomial  factor  is  removed  from  each  group 
the  polynomial  factors  in  all  the  groups  will  be  alike  in  every  respect. 

Factor,  and  test  each  result,  especially  for  signs : 

3.  am  —  an  4-  fnx  —  nx.  14.  ar  —  rs  —  ab-\-  bs. 

4.  be  —  bd -^  ex  —  dx.  15.  ar^-j-x^-f  a; -}- 1. 

5.  pq  —  px  —  rq  -\-  rx.  16.  y'  +  y*  —  3  ?/  —  3. 

6.  ay  —  by  —  ab -\- bK  17.  xr^  +  x"^ -{- a^y  +  y. 
1.  a^  —  xy  —  5x-{-5y.  IS.  2  —  n^—2n-hn\ 

8.  b^  —  bc-\-ab  —  ac.  19.  ax  —  x  —  a  +  x^. 

9.  x^-^xy-ax-ay.  20.  12a' -Sb  -  Sa^  +  2ab, 

10.  c^  — 4c  +  ac  — 4a.  21.  Sax-\-6ay  —  4:bx  —  3by. 

11.  l-m-\-n-mn.  22.  3a^-15x-{-10y -2x^y. 

12.  2x-y  +  4:Qi^-2xy.  23.  3r2^- 9  r^^^ar-Sa^ 

13.  2p-{-q  +  ()p^-{-8pq.  24.  ax— a  —  bx-\-b-'CX-\-c. 


110  FACTORING 

146.   To  factor  by  the  factor  theorem. 

Zero  multiplied  by  any  number  is  equal  to  0. 
,  Conversely,  if  a  product  is  equal  to  zero,  at  least  one  of  the 
factors  must  be  0  or  a  number  equal  to  0. 

If  5  flj  =  0,  since  5  is  not  equal  to  0,  x  must  equal  0. 

If  5  (x  —  3)  =  0,  since  5  is  not  equal  to  0,  x  must  have  such 
a  value  as  to  make  ic  —  3  equal  to  0 ;  that  is,  a;  =  3. 

If  5  (a;  —  3),  or  5  a?  —  15,  or  any  other  polynomial  in  x  reduces 
to  0  when  a;  =  3,  a;  —  3  is  a  factor  of  the  polynomial. 

Sometimes  a  polynomial  in  x  reduces  to  0  for  more  than  one 
value  of  X.     For  example,  x^  —  5  a;  -f-  6  equals  0  when  aj  =  3  and 
also  when  aj  =  2 ;  or  when  a;  —  3  =  0  and  a;  —  2  =  0.    In  this  case 
both  a;  —  3  and  a;  —  2  are  factors  of  the  polynomial. 
...  «2_5^_^6=(a;-3)(a;-2). 

/    147.    Factor  Theorem.  —  If  a  polynomial  iyi  x,  having  positive 
nntegral  exponents,  reduces  to  zero  when  r  is  substituted  for  x,  the 
polynomial  is  exactly  divisible  by  x  —  r. 
XThe  letter  r  represents  any  number  that  may  be  substituted  for  x. 

EXERCISES 

148.    1.   Factor  ar^-a;2- 4  aj  + 4. 

Solution.  —  When  x  =  1,     a;^  —  x^  —  4a-.  -f-4  =  l  —  1  —  4  +  4  =  0. 

Therefore,  a:  —  1  is  a  factor  of  the  given  polynomial. 

Dividing  x*  —  a;^  —  4  x  +  4  by  x  —  1,  the  quotient  is  found  to  be  x'^  —  4. 

By  §  134,  a;2  _  4  ^(a;  4-  2)(x  -  2). 

...  yfi  -  x^  -  ix  +  i  =(ix  -  l)(x  +  2)(ix-  2). 

Suggestions.  —  1.  Only  factors  of  the  absolute  term  of  the  polynomial 
need  be  substituted  for  x  in  seeking  factors  of  the  polynomial  of  the  form 
x  -  r,  for  if  X  -  r  is  one  factor,  the  absolute  term  of  the  polynomial  is 
the  product  of  r  and  the  absolute  term  of  the  other  factor. 

2.  In  substituting  the  factors  of  the  absolute  term,  try  them  in  order 
beginning  with  the  numerically  smallest. 

2.  Factora53-|-a;2-9a;-9. 

Suggestion.  —  When  x  =  1,  x^  +  x'^  —  9  x  —  9  =  —  16. 
Therefore,  x  —  1  is  not  a  factor  of  the  given  polynomial. 
Whenx=-1,  x^ +  x2-9x  -  9  =  0. 

Therefore,  x—  (-1),  or  x4- 1,  is  a  factor  of  the  given  polynomial. 


FACTORING  111 

Factor  by  the  factor  theorem : 

3.  13a^-6x-S.  10.  a^- 19  a; +  30. 

4.  a^-jx-^6.  11.  .x-3-67a;-126. 

5.  x'-9x--\-23x-W.  12.  m^  +  7m2+2m-40. 

6.  aj3-4a^-7  aj+lO.  13.  x^  -  25  x' -\- 60  x  -  36. 

7.  0^-6x^-9  x  +  U.  14.  a;*  +  13  ar^  —  54  a; -h  40. 

8.  a^-lla^+31a;-21.  15.  ;r* -f- 22  ar' 4- 27  a; - 50. 

9.  aj3- 10^2  ^29  a; -20.  16.  a;^-9a;3+ 21  a^+x- 30. 
17;  Factor2a:3  4-a^i/-5a;?/2  +  2/. 

Suggestion.  —  When  x  =  y, 

2  3fi-\-  x^y-bxy'^  +  2  j/S  =  2 y^  +  r/3  -  5  ys  +  2 y^  =  o. 
Therefore,  a;  —  y  is  a  factor  of  2  x^  -\-xhf  —  b  xy'^  +  2  y^. 

Factor  by  the  factor  theorem : 

18.  af»-13a;2/'^-|-122/3.  20.   a;*-9  a^/  +  12an/»-42/*. 

19.  3?-3\xy^-30f.  21.  x^ -9 o^y^ - 4: xf  ■\- 12 y\ 

MISCELLANEOUS  EXERCISES 

149.  In  the  exercises  under  the  preceding  cases,  except 
those  under  the  factor  theorem,  the  expressions  given  have 
been  completely  factored  by  one  application  of  a  single  case,  but 
frequently  it  is  necessary  to  apply  two  or  more  cases  in  suc- 
cession or  one  case  more  than  once  to  factor  the  given  expres- 
sion completely. 

Monomial  factors  should  usually  first  be  removed,  as  they 
often  disguise  a  familiar  type  form. 

1.   Factor  a:3^3aJ2_;[Oa;. 

Solution 

By  §  133,  a:8  +  3a;2_i0x  =  a;Cx2-h3ar-10) 

By  §142,  =.r(ar  +  6)(x-2). 


112  FACTORING 

2.  Factor  a;^  —  2/". 

Solution 
Writing  the  expression  as  the  difference  of  two  squares,  we  have, 

§134,  =(x3  +  2,^)(x3-2^3) 

§  136,  =(.x  +  y)  (x2  -xy  +  y"^)  {x  -  yXx^  +  xy  +  y^). 

3.  Factor  a^  —  y\ 

Solution 
By  successive  applications  of  §  134, 

a^-y»=(x'i  +  yi){Xi-y*) 

=  ix^  +  y')(x^  +  y^)(x  +y){x-  y). 
Factor  completely,  and  test  each  result : 

4.  x^-xy^,  6.    a^-h\         8.    w;*-81.  10.    ^ -1. 

5.  m  —  7n*.  7.   a^— 1.  9.   3^  — 3^.         11.   5c^—5. 

12.  4a-4a2  +  a^  16.  ?i^  +  2  71 -3  w^- 6. 

13.  2x^-^r)xy  +  2y\  17.  7^^  +  2  r' +  5  7^ -\- 10. 

14.  10  .^•2- 20^+10.  18.  18a26  +  60a52-f  50  6^. 

15.  lla-a;  — 55afl7  +  66  a;.  19.  lo^ -\- uw^ -\- vw^  +  uvw. 

20.  a;*'  —  /.  23.   4:  ax +2  aoiy^  —  4S  a.  26.   27w  +  n^ 

21.  Z^-16Z.         24.    18m=^-3m-36.  27.   64-2  a*. 

22.  Sr'-3s\      25.   y -\- 10  b'y-h  25  b*y.  28.    m^+mV. 

29.  21a2-a-10.  35.  aj'-12  a^  +  41  a;- 30. 

30.  -a2_4a  +  45.  36.  30 -m-6  m^ -\-m\ 

31.  16  a:^  _f_  20  a; -66.  37.  3  ^-^s  -  9  rs^ -j.  6r  -  3  6s. 

32.  36a;2-48a;-20.  38.  15tP-9r-n-S5tl+21ln. 

33.  a^-21a;?/2+20?/3^  39.  w*-w^-7w'^  +  w  +  6. 

34.  3a26a^-3a26a;-6a'''6.  40.  m*- 15  m2  +  10m  +  24. 


FACTORING  113 


SPECIAL   APPLICATIONS    AND    DEVICES 

150.  The  method  of  factoring  by  grouping  the  terms  of  an 
expression  in  certain  ways  is  very  important.  Polynomials 
may  often  be  arranged  in  some  one  of  the  type  forms  already 
studied,  and  even  many  of  these  types  themselves  may  be 
factored  by  grouping  to  show  a  common  polynomial  factor. 

151.  To  factor  a  polynomial  that  may  be  grouped  to  form  the 
difference  of  two  squares. 

Just  as,  §  134,      a^  -62  =  (fl  +  6)  (a  -b), 
so  a^  -  (6  +  c)-  =  [a  +  (6  +  c)]la  -  (ft  -h  c)] 

=  (a  +  6  H-  c)(a  —  b  —  c). 

EXERCISES 

152.  Factor: 

1.  (a  +  6)2_c2.  3.   x'-(y-z)\'         5.    (a-bf-i^, 

2.  )^-(s  +  ty.  4.    (l-\-my-n\  6.    l-{v-\-w)\ 
7.    ¥siGt0TX*-(3x''-2yy-. 

Solution 

a:4_(3a;2_2?/)2  =  [x-2+  (S  x"^  -  2  y)][x^ -  (3  x^  -  2  y)} 
=  (x^  +  Sx^  -2y)(x^  -  Sx^  +  2  y) 
=  (4x2-2y)(2y-2x-2) 
=  2{2x^-y)2{y-x^) 
=  4(2x^-y)iy-x'^). 
Test.  —  When  x  =  2  and  y  =  3, 
a^_  (3  a;2_  2  yy^  =  2*  -  (3  •  2^  -  2  •  3)2  =  16  -  (12  -  6)2  =  16  -  36  =  -  20, 
and  4(2  x^  -  y){y  -  x2)  =  4(2  ■  22  -  3)  (3  -  22)  =  4(8  -  3) (3  -  4)  =  -  20. 

Factor,  and  test  each  result : 

8.  4c2-(6  +  c)2.  12.   49r2-(5r-4s)2. 

9.  (2a-irhY-b\  13.    ?;Qz^ -  {3z-l yf. 

10.  9  «2 -(2^-5)2.  14.    (6?()-3A')2-64A:2^ 

11.  25a2-(ft4-c)-.  15.    (3m4-8?0'-16m2. 

milne's  1st  yr.  alg. —  8 


114  FACTORING 

16.  Factor  a^  -|-  4  -  c*  -  4  a. 

Sl'ggestion. — The  given  expression  contains  three  square  terms  and  a 
term  that  is  not  a  square.  The  latter  may  be  the  middle  term  of  a  trino- 
mial square.  If  so,  it  contains  as  factors  the  square  roots  of  two  of  the 
square  terms  and  these  are  the  other  terms  of  the  trinomial  square. 

Then,  arranging  and  grouping  terms,  we  have 

^2  +  4  -  c2  -  4  a  =  (a2  -  4  a  +  4)  -  c2  =  (a  -  2)2  -  c2. 

Note.  —  It  will  be  observed  that  the  term  that  is  not  a  perfect  mono- 
mial square  furnishes  a  key  to  the  grouping. 

Factor,  and  test  each  result : 

17.  w'-2ax  +  x'-n\  21.  c" -  a"^ -b^ -2ab. 

18.  h^-{-2by  +  y^-n\  22.  b' -x"  -  y^ +  2xy. 

19.  1  —  4  g  -f  4  ^2  _  ^2  23.  4  c^  —  oc^  —  f  —  2  xy. 

20.  r2-2ra  +  aj2-16^2  24.  Oa^  _  6a6  +  6^  _  4^2^ 

153.  The  principle  by  which  the  difference  of  two  squares  is 
factored  may  be  extended  to  expressions  that  may  be  written 
as  the  difference  of  two  squares  by  adding  and  subtracting  the 
same  monomial  perfect  square. 

EXERCISES 

154.  1.    Factor  a^  +  a^ft^  +  6^ 

Suggestion.  —  Since  a^  +  a'^h'^ -{■  h^  lacks    -\- a'^h'^  of  being  a  perfect 
square,  and  since  the  value  of  the  polynomial  will  not  be  changed  by  add- 
ing a2&2  and  also  subtracting  a%^^  the  polynomial  may  be  written 
a*  +  2  am  -H  6*  -  a262,  or  {a^  +  62)2  _  ^^252. 

2.    Factor  a;<  -  21  a^  +  36. 

Suggestion.  — This  expression  has  two  square  terms,  but  in  order  that 
it  shall  be  a  perfect  trinomial  square  it  must  fulfill  another  condition, 
namely  (§  140),  the  other  term  must  be  numerically  equal  to  twice  the 
product  of  the  square  roots  of  the  terms  that  are  squares ;  that  is,  the 
middle  term  must  be  either  -t-  12  x"^  or  —  12  x\ 

Hence,  the  number  to  be  added  and  subtracted  is  either  33  a;2  or  9  a;2, 
but  the  former  will  not  give  the  difference  ^1  two  squares,  for  33  x"^  is  not 
a  perfect  square ;  then,  9  x2  is  the  number  to  be  added  and  subtracted, 
giving 

a;4_  12x2  +  36-9x2,  or  (x2  -  6)2  -  9x2. 


FACTORING  115 

Separate  into  prime  factors,  and  test  each  result : 

3.  x*-{-  x^f-  4-  2/*-  8.   a;^  +  a:^  +  1. 

4.  a^+a'h^  +  h\  9.    /i^  +  n^  +  1. 

5.  9x^  +  20Q^y^-J^lQy\  10.    16 a;*  +  4 it-^i/^  +  ?/*. 

6.  4a*  +  lla262^-9  6^  11.   25  a^  -  14  a^ft^  ^  fts^ 

7.  16  a*- 17  aV  4-0^.  12.   9  a*  +  26  a^d^  +  25  6^ 

13.  Factor  a* +  4. 

Suggestion,      a^  +  4  =  a*  +  4  a2  ^.  4  _  4  ^a  _  (aj2  _^  2)2  -  4  a^. 

Factor  completely,  and  test  each  result : 

14.  a*  +  4  h\  16.   a:^  -  16.  18.   m*  +  4  mn* 

15.  m*4-64.  17.   4  a* +  81.  19.   a^y^ -\- A^  xy\ - 

155.  The  method  of  factoring  by  grouping  to  show  a  common 
polynomial  factor  applied  to  cases  already  solved  by  other  methods. 

The  student  has  learned  how  to  factor  several  type  forms 
by  special  methods.  He  will  now  see  how  many  of  these  forms 
may  be  factored  by  the  method  of  §  145,  which  is  of  impor- 
tance because  its  application  is  so  general. 

EXERCISES 

156.  Tlie  forms  a^-\-2ah  +  h^  and  a?-2ab  +  b\ 

1.  Factor  ar*  +  2  a?!/ 4- y^ ;  also  9  a%2_  6^^1  +  1. 

Solution  Solution 

x^  +  2xy  +  y'^  9  a^m^  -  6  am  +  I 

=  x^-\-xy  +  xy  +  y^  =9  a^m^  -  3  am  -  3  am  +  1. 

=  x(x -^  y)  +  y  (x -\- y)  =  Sam(Sam-  l)  —  {Sam—  1) 

=  (x  +  y)(x-hy).  =(3aw-l)(3am-l). 

Factor  by  separating  and  grouping ;  test  each  result : 

2.  ?2  +  i4;4.49.  4.   4ar'  +  12a;2/4-92/*. 

3.  /-^-ISr-fSl.  5.   16a=^-24a6  +  96l 


116 


FACTORING 


The  form  a^-h\ 
6.    Factor  x^-y^-,  also  9  r^  -  4  s^. 


Solution 

=  x(x-y)-[-y(x-y) 
=  (x  +  y)(x^y). 


Solution 

9r2-4s2 
=  9r2  -  6  rs  +  6 rs  -  4 s2 

=  3r(3r-2s)+2s(3r-2s) 
=  (3r  +  2s)(3r-2s). 


Factor  by  grouping ;  test  each  result 


7.    a' 


z\ 


9.    n^-\, 
8.    x^-\.  10.    9a^-25. 

2%e  form  oi^ -\- px  •\- q. 
13.   Factor  a;^  +  8  a;  +  15 ;  also  a^  -  2  a;  — 15. 


11.  3622-25i;2, 

12.  49n2-100;2. 


Solution 

=  a;2  +  5ic+3a;  +  15 
=  a:(x+5)  +  3(x  +  5) 


Solution 

a;2_2a;-15 
=  a;2_5a;  +  3a;_  15 

=  a;(a:-5)+3(ic-5) 
=  (a;  +  3)(x-5). 


Factor  by  separating  and  grouping ;  test  each  result : 

14.  a^  +  12a;  +  20.  16.   2/^+8?/-20. 

15.  ?7i2_^9m  +  18.  17.    ^2-57^-14. 

The  form  aa? -{- hx '\'  c. 

18.   Factor  2  a^H- 11  aj-j- 12;  also  2  a;'^  +  a;  - 15. 


Solution 

2a:2+lla;  +  12 
=  2x2 +  8  a: +  3  a; +  12 
=  2a;(a;  +  4)  +  3(x  +  4) 
=  (2x  +  3)(x4-4). 


Solution 

2x2  +  a;-15 
=  2x2  +  6ic-5x-  15 

=  2a;(x+3)-  5(a;  +  3) 
=  (2ic-5)(x  +  3). 


Factor  by  separating  and  grouping ;  test  each  result : 

19.  2ar^  +  3a;  +  l.  21.   3m^-{-5m-22. 

20.  92/'-h2l2/  +  10.  '  22.   10  7^-3rs-lS  s\ 


FACTORING  117 

Factor  by  separating  and  grouping : 

23.  12n2  +  31n  +  9.  25.    20 x'-^-lSxy-lBy^. 

24.  5^24.482-77.  26.    Ua'-23ab-30b\ 

REVIEW   OF    FACTORING 

157.    Summary  of  Cases.  —  In  the  previous  pages  the  student 
has  learned  to  factor  expressions  of  the  following  types : 

Monomial  Factors 

I.   Of  monomials ;  as  a-b'c.  (§  129) 

II.   Of  expressions  whose  terms  have  a  common  factor ;  as 

/7jr  H"  /»/  H-  nz.  (§  133) 

'  Binomials 

III.  Difference.of  two  squares ;  as 

a--b'.  (§§  134,161,155) 

IV.  Sum  or  difference  of  two  cubes ;  as 

a'  +  b'ova^-b'.  (§136) 

V.    Sum  or  difference  of  same  odd  powers ;  as 

a"  4-  b"  or  a"  -  6"  (when  n  is  odd).  (§  137) 

Trinomials 

VI.   That  are  perfect  squares ;  as 

a'  +  2ab  +  b'  and  a'  -  2  ab  4-  b'.  (§§  138,  155) 

VII.   Of  the  form          x-+px-\-  q.  (§§  142,  155) 

VIII.   Of  the  form         ax'  -hbx-\-c.  (§  §  144,  155) 

Polynomfals 

IX.    Whose  terms  may  be  grouped  to  show  a  common  poly- 
nomial factor ;  as 

ax  +  ay-\-bx  +  by.  (§  §  145,  155) 

X.    Having  binomial  factors  (Factor  Theorem).         (§  146) 


118  FACTORING 

158.  General  Directions  for   Factoring   Polynomials.  —  1.  Re- 
move monomial  factors  if  there  are  any. 

2.  Then  endeavor  to  bring  the  polynomial  under  some  one  of 
the  cases  II-IX. 

3.  Whe7i  other  methods  fail,  try  the  factor  theorem. 

4.  Resolve  into  prime  factors. 

Each  factor  should  be  divided  out  of  the  given  expression  as  soon  as 
found  in  order  to  simplify  the  discovery  of  the  remaining  factors. 

EXERCISES 

159.  Factor,  and  test  each  result: 

1.  2/4-1.  8.  l  +  aj'2  15  8-27aV. 

2.  1— a^.  9.  y-a^y.  16.  32«  — 2a5^. 

3.  x^'^-1.  10.  x^y-f.    ^  17.  6  6^  +  24. 

4.  x^-1.  11.  a^^-ab^  I  18.  a^  +  27al 

5.  a -a'.  12.  64.-2  f.  19.  6^-196. 

6.  6^  +  6.  13.  In' -In.  20.  450  -  2  a^. 

7.  j94-t-4.  14.  4ic*  — 4  a;.  21.  Ty^  — 175. 

22.  a^-xy-lS2y\  32.  a:^-ax-72a^. 

23.  aar^  —  3  aa;  —  4  a.  33.  n^  — an— 90  a^. 

24.  a!3-l-5a^-6a;.  34.  a-b^ -{■  ab  —  56. 

25.  3a^  +  30a;  +  27.  35.  a^n  _  2  a"6«  +  6^ 

26.  128  a^- 250  a^.  36.  25  a;^  +  60  a;^/ +  36 /. 

27.  6  a^- 19  a; +  15.  37.  6  aa^ -\- 5  axy —6  ay\ 

28.  a;2«  +  2a;"?/P  +  2/^^.  38.  169  a;^  -  26  aa;«  +  a V. 

29.  7a;2_77a^_84  2/2.  39.  a*c^  +  a^b^c^ -{- b\ 

30.  2/'-25  2/a;  +  136a;2.  40.  16  a:^ -f  4  a^y  +  ^^ 

31.  9:x^-24.xy-\-16y\  41.  6*c  - 13  6^c  +  42  c. 


FACTORING  119 

Factor,  and  test  each  result : 

42.  9'j^-{-21x-{-10.  55.  b'  +  by  +  y*. 

43.  5  a^  -  26  xy -\- 5  y\  66.  xy  —  Sy  +  x  —  Sf. 

44.  2/^-hl6a2/  — 36  al  57.  ast?  —  axy  —  ax -\- ay. 

45.  8a2-21a6-962.  53.  9  c2_aj2_y2_^2a^. 

46.  9  a:^  — 15  ic  —  50.  59.  ir^  —  a-a;  —  4  fe^a;  _  4  a&a?. 

47.  30a^-37a;-77.  60.  b&  -  9  a'b  -  b^  -  6  ab\ 

48.  207^  +  28^2 _,_eg^  61  a62_4a«-12a2c-9ac2. 

49.  a2^62_^_2a5.  62.  a^- ca;+ 2da;-2cd. 

50.  aiB^-f  10  aa;- 39  a.  63.  a^y  +  4  a^2/ -  31  a;y  -  70  y. 

51.  n^  +  n'a'b'^a'b^*  64.  ar^  -  3  aa;  +  4  6a;  -  12  a6. 

52.  aV  +  aV  +  a*.  65.  aa»  -  9  aar' +  26  aa;  -  24  a. 

53.  a^"*- 16  a*" -17.  66.  12aa;-86a;-9  ay +  6  ft?/. 

54.  aV-4aa;4-3.  67.  25 af- 9  2^-24^2-16  22. 

68.  2bH-^ab^  +  2btx-^dbx, 

69.  a:^y-f-14ar'y  +  43ary-f  3O2/. 

70.  arV  - 15  a;22^  + 38^2/ -24  2/. 

71.  ab3i?-\-^(iboi?  —  abx  —  ^ab. 

72.  3  bmx  +  2  6m  —  3  anx  —2  an. 

73.  20  aa;3  _  28  aa;2  4- 5  a^a;- 7  a^. 

74.  (a +  6)3  —  1.  79.  a:^  —  a.-2  —  a;*  +  a;^. 

75.  a^  —  2  a^-^-l.  80.  a^  —  xy  —  a^y -\- y^. 

76.  6^-462  +  8.  81.  12  a;3^3  aj2_8^_2. 

77.  3a;^  +  96a;.  82.  2  a;^  _j_  ^Lq  ^  _j.  ^a,  _,_  5  ^ 

78.  8a;*-6a;2-36.  83.  m^ -\- m^ - wm - mn^. 

84.  Factor  16  4-  5  a;  — 11  o^  by  the  factor  theorem. 

85.  Factor  a;^  —  6  6ar  +  12  6^a;  —  8  6^  by  the  factor  theorem. 


120  FACTORING 


EQUATIONS   SOLVED   BY   FACTORING 

160.  Equations  thus  far  solved  have  been  such  as  involved, 
when  in  simplest  form,  only  the  first  power  of  the  unknown 
number. 

Such  equations  are  called  simple  equations. 

161.  A  valuable  application  of  factoring  is  found  in  the 
solution  of  equations  that  involve  the  unknown  number  in 
powers  higher  than  the  first. 

An  equation  that  in  simplest  form  involves  the  second,  but 
no  higher,  power  of  the  unknown  number  is  called  a  quadratic 
equation. 

Thus,  x2  =  4  and  x'^  +  2  a;  +  1  =  0  are  quadratic  equations  ;  but  x^-\-^x 
=  ic2  +  4  is  a  simple  equation,  for  in  its  simplest  form,  3  x  =  4,  it  has  only 
the  first  power  of  x. 

162.  An  equation  that  contains  a  higher  power  of  the 
unknown  number  than  the  second  is  called  a  higher  equation. 

163.  To  solve  quadratic  equations  by  factoring. 

EXERCISES 

1.   Find  the  values  of  x  that  satisfy  a^  + 1  =  10. 
Solution 

X2  +  1  =  10.  (1) 

Transposing  so  that  all  terms  are  in  the  first  member  and  uniting  terms, 

a;2  _  9  =  0.  (2) 

Factoring  the  first  member,  §  134, 

(x-3)(x  +  3)  =  0.  (3) 

Since  the  product  of  the  two  factors  is  0,  one  of  them  must  equal  0 ; 
that  is,  the  equation  is  satisfied  for  any  value  of  x  that  will  make  either 
factor  equal  to  @. 

Ifx-3  =  0,  x  =  3;  ifx  +  3  =  0,  a;=-3. 

Hence,  the  values  of  x  that  satisfy  (3)  and  therefore  (1)  are  3  or  —  3. 

Verification.  —  When  x  =  3,  (1)  becomes  9  +  1  =  10,  or  10  =  10. 

When  x  =  -  3,  (1)  becomes  9  +  1  =  10,  or  10  =  10. 


FACTORING  121 

Solve,  and  verify  results : 

2.  i»2 4-3  =  28.  6.  or' 4-3  =  84. 

3.  a;^4-l  =  50.  7.  ar^-24  =  120. 

4.  x'-5  =  59.  8.  a:2  +  ii=,i80. 

5.  a:2_7^29.  9.  a:2_ii==iio. 

10.  Solve  07^  4- 4  a;  =  45. 

Solution 

Transposing,  x=*  -H  4  a;  —  45  =  0. 

Factoring,  §  142,  (x  -  5)  (x  +  9)  =  0. 

Hence,  x—  5=:0orx-}-9  =  0; 
whence,  x  =  6  or  —  9. 

Solve,  and  verify  results : 

11.  a^-6x  =  ^0.  16.  2/2  4-42  =  132/. 

12.  a^-8a;  =  48.  17.  t^-\-63  =  mt 

13.  ar'  -  5  a;  =  —  4.  18.  v^  —  60  =  11  v. 

14.  a;2  4-4a;4-3  =  0.  19.  ar^-7a;  =  18. 

15.  r2  4-6r4-8  =  0.  20.  a?^ 4- 10 a;  =  56. 

21.  Solve  6ar^4-5a;-21=0. 

Solution 
6x-2  +  6x-21  =  0. 
Factoring,  §  144,        (2  x  -  3)  (3 x  4-  7)  =  0. 
Hence,  2x-3=0  or  3x4-7=0; 

whence,  x  =  |  or  —  J. 

Solve,  and  verify  results  : 

22.  3a;^4-2a;-l=0.  27.  2v^-9r-3^  =  0. 

23.  5a^4-4a;-l=0.  28.  6y^ - 22 y -^20  =  0. 

24.  Sf-\-y-10  =  0.  29.  622_ll2_21  =  0. 

25.  7a:2  4.6«-l=0.  30.  4ar^- 15a;4- 14  =  0. 

26.  23^*4- 9a;-18  =  0.  31.  53^^-483;-- 20  =  0. 


122  FACTORING 

Solve,  and  verify  results : 

32.  a.-2-21  =  4.  41.  32-ar^  =  28. 

33.  i»2-56  =  8.  -       42.  65-0^=16. 

34.  a^-9aj  =  36.  43.  a.-^ -f- a- -  132  =  0. 

35.  a?-{-llx  =  26.  44.  S2  =  4:W-{-w\ 

36.  a^-12a;  =  45.  45.  3s  =  88-s2. 

37.  2/2_i52/  =  54.  46.  160  =  a^-6a;. 

38.  /-2l2/  =  46.  47.  42/  =  /-192. 

39.  3/-42/-4  =  0.  48.  3a^  +  13a-30  =  0. 

40.  42/'  +  92/-9  =  0.  49.  4a^  +  13a;- 12  =0. 

50.  (2»  +  3)(2a;-5)-(3a;-l)(a;-2)  =  l. 

51.  (2a;-6)(3aj-2)-(5aj-9)(a;-2)  =  4. 
Other  methods  of  solving  quadratics  will  be  given  in  §§  336-348. 

164.  To  solve  higher  equations  by  factoring. 

Any  higher  equation  may  be  solved  by  the  method  just 
given  for  quadratic  equations,  whenever  the  expression  ob- 
tained by  transposing  all  of  its  terms  to  one  member  is 
factorable. 

EXERCISES 

165.  1.    Solve  the  equation  a^  —  2a^  —  5a;  +  6  =  0. 

Solution 

Tactormg,  §  146,  (x  -l)(x-  3)  (a;  +  2)  =  0. 

Hence,  a:-l  =  0ora;-3=0oric  +  2  =  0; 

whence,  aj  =  1  or  3  or  —  2. 

2.  x^-15x'-{-71x-105  =  0.         4.    a^-12x-\-16  =  0. 

3.  iB3  4-10aj24-ll«-70  =  0.  5.   a^- 19aj-30  =  0. 

6.  a;*-|-a^-21a^-a;  +  20  =  0. 

7.  a;*-7a^-f a;24-63a;-90  =  0. 

8.  x^-10x^  +  35x'-50x  +  24.  =  0. 


FACTORING  123 

Problems 

166.  1.  A  sealing  fleet  carries  4000  men,  and  the  number  of 
men  on  each  ship  is  40  less  than  8  times  the  number  of  ships. 
Find  the  number  of  ships  and  the  number  of  men  on  each  ship. 

Solution 

Let  X  =  the  number  of  ships  in  the  fleet. 

Then,  8  a;  —  40  =  the  number  of  men  on  each  ship, 

and  x(S  x  —  40)=  the  total  number  of  men  with  the  fleet. 

.-.  a;(8  X  -  40)  =  4000. 

Expanding,  dividing  by  8,  and  transposing, 
a;2  _  5  a;  _  .500  =  0. 

Factoring,     (x  -  26)  (x  +  20)  =  0. 

Hence,  x  -  25  =  0  or  x  -|-  20  =  0  ; 

whence,  x  =  26  or  —  20, 

and  8  X  -  40  =  160  or  -  200. 

The  second  value  of  x  and  of  8  x  —  40  is  evidently  inadmissible,  since 
neither  the  number  of  ships  nor  the  number  of  men  on  a  ship  can  be 
negative. 

Hence,  there  are  26  ships  in  the  fleet,  and  160  men  on  a  ship. 

Solve  the  following  problems  and  verify  (§  125)  each 
solution : 

2.  The  gold  mined  in  a  recent  year  would  fill  a  square  room, 
the  height  of  which  is  1  foot  less  than  its  length.  If  the  area 
of  one  wall  is  90  square  feet,  find  the  dimensions  of  the  room. 

3.  Certain  wooden  paving  blocks  are  twice  as  long  as  they  are 
wide  and  the  thickness  of  each  is  4  inches.  Find  the  length 
and  width,  if  the  volume  of  each  block  is  128  cubic  inches. 

4.  A  rectangular  swimming  tank  on  board  a  ship  is  3  times 
as  long  as  it  is  wide.  If  it  were  divided  into  3  square  tanks, 
the  area  of  each  would  be  225  square  feet.  Find  the  dimen- 
sions of  the  tank. 

5.  A  man  bought  as  many  tons  of  crude  borax  as  it  is  worth 
dollars  a  ton,  and  crude  borax  is  worth  -f  as  much  as  refined 
borax.  If  the  same  amount  of  refined  borax  would  be  worth 
{$  2800,  find  the  value  of  crude  borax  a  ton. 


124  FACTORING 

6.  A  farmer  keeps  his  chickens  in  a  rectangular  lot  that  is 
4  times  as  long  as  it  is  wide.  If  its  area  is  2500  square  feet, 
find  its  length  and  width. 

7.  At  a  luncheon  with  Menelik  of  Abyssinia  there  was  a 
pile  of  bread  containing  448  cubic  feet.  Its  height  was  twice 
its  width  and  its  length  was  14  feet.  Find  its  height  and 
width. 

8.  A  large  rectangular  freight  station  in  Atlanta,  Georgia, 
covers  an  area  of  41,750  square  feet.  If  the  length  is  16.7 
times  the  width,  what  is  the  width  of  the  station  ? 

9.  An  automobilist  paid  $  3.60  for  gasoline.  If  the  num- 
ber of  cents  he  paid  per  gallon  was  2  less  than  the  number  of 
gallons  he  bought,  find  how  many  gallons  he  bought  and  the 
price  per  gallon. 

10.  St.  Louis  has  the  largest  steam  whistle  in  the  world. 
The  number  of  times  it  is  blown  each  day  is.  3  more  than  the 
number  of  dollars  it  costs  to  blow  it  once.  How  many  times  is 
it  blown  a  day,  if  the  cost  for  12  days  is  $48  ? 

11.  In  the  Panama  Canal  Zone  a  washerwoman  washed  as 
.  many  dozen  pieces  as  she  received  dollars  a  dozen  for  her  labor. 

If  she  had  washed  2  dozen  more,  she  would  have  received  $  15. 
How  much  did  she  receive  a  dozen  ? 

12.  The  number  of  pounds  of  duck  feathers  that  a  man 
bought  was  the  ^ame  as  the  number  of  cents  that  he  paid  a 
pound  for  them.  If  he  had  bought  10  pounds  more,  they  would 
have  cost  $  20.     How  many  pounds  did  he  buy  ? 

13.  The  area  of  one  of  the  largest  photographic  prints  ever 
made  is  180  square  feet.  Its  dimensions  are  18  times  those  of 
the  picture  from  which  it  was  enlarged.  Find  the  dimensions 
of  the  picture,  if  its  length  is  2  inches  greater  than  its  width. 

14.  The  cages  holding  canaries  imported  into  this  country 
are  arranged  in  rows  in  crates.  -  The  number  of  rows  in  a  crate 
is  2  less  than  5  times  the  number  of  cages  in  a  row.  If  there 
are  231  cages  in  a  crate,  how  many  rows  are  there  ? 


FRACTIONS 


167.  lu  algebra,  an  indicated  division  is  called  a  fraction. 

The  fraction  -  means  a  -i-  b  and  is  read  '  a  divided  by  b.' 

h 

168.  The  dividend,  written  above  a  line,  is  the  numerator; 
the  divisor,  written  below  the  line,  is  the  denominator ;  the  nu- 
merator and  denominator  are  called  the  terms  of  the  fraction. 

An  arithmetical  fraction  also  indicates  division,  but  the  arithmetical 
notion  is  that  a  fraction  is  one  or  more  of  the  equal  parts  of  a  unit ;  that 
is,  in  arithmetic,  the  terms  of  a  fraction  are  positive  integers,  while  in 
algebra  they  may  be  any  numbers  whatever. 

169.  The  student  will  find  no  difficulty  with  algebraic  frac- 
tions, if  he  will  bear  in  mind  that  they  are  essentially  the  same 
as  the  fractions  he  has  met  in  arithmetic.  He  will  have  occa- 
sion to  change  fractions  to  higher  or  lower  terms ;  to  write  in- 
tegral and  mixed  expressions  in  fractional  form ;  to  change 
fractions  to  integers  or  mixed  numbers ;  to  add,  subtract,  mul- 
tiply, and  divide  with  algebraic  fractions  just  as  he  has  learned 
to  do  with  arithmetical  fractions,  except  that  signs  must  be 
considered  in  dealing  with  positive  and  negative  numbers. 

Signs  in  Fractions 

170.  There  are  three  signs  to  be  considered  in  connection 
with  a  fraction;  namely,  the  sign  of  the  numerator,  the  sign  of 
the  denominator,  and  the  sign  written  before  the  dividing  line, 
called  the  sign  of  the  fraction. 

In  —  ^  the  sign  of  the  fraction  is  — ,  while  the  signs  of  its  terms  are  -h . 
oz 

126 


126  FRACTIONS 

171.    An  expression  like  — —  indicates  a  process  in  division, 

—  b 

in  which  the  quotient  is  to  be  found  by  dividing  a  by  6  and 

prefixing  the  sign  according  to  the  law  of  signs  in  division ; 

that  is, 


-a_     a 

+  a  a 
+  6      "^6 

—  a         a 
+  b~    ,6' 

+  a_  a 
-b          b 

By  observing  the  above  fractions  and  their  values  the  follow- 
ing principles  may  be  deduced : 

172.  Principles. — 1.  TJie  signs  of  both  the  numerator  and 
the  denominator  of  a  fraction  may  be  changed  without  changing 
the  sign  of  the  fraction. 

2.  The  sign  of  either  the  7iumerator  or  the  denominator  of  a 
fraction  may  be  changed,  provided  the  sign  of  the  fraction  is 
changed. 

When  either  the  numerator  or  the  denominator  is  a  polynomial,  its  sign 
is  changed  by  changing  the  sign  of  each  of  its  terms.  Thus,  the  sign  of 
a  —  6  is  changed  by  writing  it  —  a  +  6,  or  &  —  a. 

EXERCISES 

173.  Keduce  to  fractions  having  positive  numbers  in  both 
terms :  x 

1    ZL?.      *^.   ±JlLi.       5       (^a-b^      7        -2-m 
-4*         '-    2x  ^,f;s'         c  +  d    '         '  2  +  n 

2.   A..        4.      -^'    .        6.   -■     -^     .       S,.  -Ha  +  b)^ 
-5  -b-y  p-a-y    ^r^Mj-x-fy) 

174.  In  accordance  with  §  80,     "  /     K  ff^— /  -J^ y 

Principles. — 3.    The  sign  of  either  term  of  a  fra^^i^H^' 
changed  by  changing  the  signs  of  an  odd  number  of  its  factors. 

4.  The  sign  of  either  term  of  a  fraction  is  not  changed  by 
changing  the  signs  of  an  even  7iumber  of  its  factor's. 


^^  ^     _ 


/ 


FRACTIONS  127 


EXERCISES 

175.   1.   Show  that  (°  -  '')  (f  -'1  =  ("  -  ^K«  -  '^ . 

(c  —  a)  (6  —  c)      (a  —  c)(6  —  c) 

SoLUTiox  OR  Proof 

Changing  (d  —  c)  to  (c  —  d)  changes  the  sign  of  one  factor  of  the  nu- 
merator and  therefore  changes  the  sign  of  the  numerator  (Prin.  3). 

Similarly,  changing  (c  —  a)  to  (a  —  c)  changes  the  sign  of  the  denomi- 
nator (Prin.  3) . 

We  have  changed  the  signs  of  both  terms  of  the  fraction.  Therefore, 
the  sign  of  the  fraction  is  not  affected  (Prin.  1). 

2.  Show  that  {b-a)(d-c)  ^  _~ia-b){c-d) . 

(c-b)(a-c)  (h-c)(a-c) 

Solution  or  Proof 

Changing  the  signs  of  two  factors  of  the  numerator  does  not  change  the 
sign  of  the  numerator  (Prin.  4). 

Changing  the  sign  of  one  factor  of  the  denominator  changes  the  sign 
of  the  denominator  (Prin.  3). 

Since  we  have  changed  the  sign  of  only  one  term  of  the  fraction,  we 
must  change  the  sign  of  the  fraction  (Prin.  2). 

3.  Show  that  -^ —  may  be  properly  changed  to 


h—a  a—b 


4.  From derive  by  proper  steps. 

6-a  +  c  a-b-c       ■       ^  ^ 

5.  Prove  that  — ^= —:  that  -     ^  ^ 


6.   Prove  that 


1  —X         x  —  1  4  —  ar^     ar^  —  4 

(b-a)(c-b)      (a-b)(b-c)' 


7.  Prove  that  (m-n)(m4-n)^     -m^  +  n^     . 

(a  —  c)(b  —  a)        (a  —  c)(a  —  b) 

8.  Prove  that     {a-b)(b-aj-_c)_ ^    (a-b)(a-b-c)    , 

(y-x)(z-y)(z-x)      (x-y)(y-z)(x-z) 


128  FRACTIONS 


REDUCTION  OF   FRACTIONS 

176.  The  process  of  changing  the  form  of  an  expression 
without  changing  its  value  is  called  reduction. 

177.  An  expression,  some  of  whose  terms  are  integral  and 
some  fractional,  is  called  a  mixed  number,  or  a  mixed  expression. 

n  h        X^  Cfi  1     ' 

Thus,  a , 2  H — ,  and  a  —  h  -\ are  mixed  expressions. 

c        a2  a;2  ab 

178.  To  reduce  a  fraction  to  an  integer  or  a  mixed  expression. 

This  change  in  form  is  made  in  algebra  in  the  same  manner 
as  in  arithmetic. 

EXERCISES 

179.  1.    Reduce  to  a  mixed  number :   -;  ^^i±^. 

4  X 

PROCESS  PROCESS 

15  =  13^4  =  3 +  -  =  3i.  ^^^±^  =  (aa.4-6)-^aj  =  a+-- 

4  4  .       a;  x 

Explanation.  —  Since  a  fraction  is  an  indicated  division,  by  perform- 
ing the  division  indicated  the  fraction  is  changed  into  the  form  of  a  mixed 
number. 


Reduce  to  an  integral  or  a  mixed  expression : 

9  '      \hxy  a 

136*  '  9c        *  *  6a; 

8.   Reduce  ^^^^ — a  —a-\-     ^^  ^  mixed  number. 


Solution.      «^-3a^-a  +  l  ^  «  _  3  4.  -  «^  +  1  =  ^  _  3  _  o^li . 

Note.  —  It  is  not  necessary  to  write  the  step  («8  —  3  a^  —  a  +  1)  -^  a^. 
The  division  should  be  continued  until  the  undivided  part  of  the  numera- 
tor no  longer  contains  the  denominator. 


a 

+  3 

4  a^  4-12 

a3_a^4-34 

2a3  +  5 

a2  +  3a& 

-5&2-f6c 

a- 

-26 

^-Ix"- 

-  4  a;  -H  40 

FRACTIONS  129 

Reduce  to  an  integral  or  a  mixed  expression : 

^    4ar'-8ic2-f2a;-l  ,_     a^  +  O  a^^  24  a -f  18 

9.    — - — — •  ly. 

2a; 

10    CLb  —  bc  —  cd  +  d^  2^ 

6 

, ,     a  V  —  aa;^  —  a;  -- 1  oi 

11. ^*  /6l. 

ax 

12     ^ZI^nl?.  22. 

a;-4  x^-3 

'  ^     x^-2xy-f  oo     «'  +  3a^6-a5^  +  a6 

lo.      = •  Ao. ;  • 

x  —  y  a^  +  h 

2xy  '  x-\-y 

,,     a^-6x'-\-Ux-9  ^^     ar^-3ar-^'*4-5af*-3 

15. •  *o. • 

x-2  a~-l 

'^-  — ^33 ^^-  — r=^7 

,^     a;3_^5^2_^33._g  :x?-\-x'y-y''-xf-Z 

17.     — •  -o/. • 

a;  -f  2  x  —  y 

.T^  +  2a:^-a.-^-f5  3  a;y-12  a:^4- 2  y^- 3 

ar'  +  ar^  *  "  3x^  +  2 

180.  To  reduce  a  fraction  to  its  lowest  terms. 

T    4.  3      9  9       3 

Just  as-  4  =  1^°'^  12  =  4'        ■ 

SO  -  =  —  or  —  =  -  •     1  hat  is, 

h      hm        bm      b 

181.  Principle.  —  Multiplying  or  dividing  both  terms  of  a  frac- 
tion by  the  same  number  does  not  change  the  value  of  the  fraction. 

182.  A  fraction  is  in  its  lowest  terms  when  its  terms  have 
no  common  factor. 

MILKENS    IST    YR.    ALG. — 9 


130 


FRACTIONS 


183.  The  product  of  all  the  common  prime  factors  of  two  or 
more  numbers  is  called  their  highest  common  factor  (h.  c.  1). 

Thus,  to  find  the  h.c.f.  of  two  expressions,  as  9  ax^y -^  S  bx^y  Siud 
6 x*y^^6  xV»  we  first  factor  them.  Since  9  ax^y-\-Sbx^y=S  ■  x^  •  y  (^Sa-{-h) 
and  6  x:^y^  —  6  a;V  =  ^  -  ^  ■  ^'^  -  y^  (x  +  y){x  —  y),  their  common  factors 
are  3,  x,  x,  and  y  ;  hence,  their  h.  c.  f.  =  3  x'^y.     (See  also  §§  407-411.) 

Note.  —  The  number  of  literal  factors  in  a  term  determines  its  degree, 
and  the  term  of  an  expression  that  has  the  greatest  number  of  literal 
factors  determines  the  degree  of  the  expression.  Thus,  2  a^h  is  of  the  third 
degree  and  is  higher  than  5  ah,  which  is  of  the  second  degree;  the  expres- 
sion 2  ^25  _|.  5  qJ)^  then,  is  of  the  third  degree. 

EXERCISES 

184.  1.   Reduce  to  lowest  terms  :    — ;  ?L^^. 

24'    SOa^xz  ^ 


PROCESS 

20 

2.2.5 

5 

24 

2.2.6 

6 

PROCESS 

21  g  Vy  ^3-7  a  Vy  ^  7  xy 
3(}a?xz      ^' 10  a^xz     10  az 

Explanation.  — Since  a  fraction  is  in  its  lowest  terms  when  its  terms 
have  no  common  factor,  a  fraction  may  be  reduced  to  its  lowest  terms  by- 
removing  in  succession  all  common  factors  of  its  numerator  and  denomi- 
nator ;  or  by  dividing  the  terms  by  their  highest  common  factor. 


Reduce  to  lowest  terms 


2     ^^ 


awrrC 

h^xy^ 


6. 


16  m^nxh^ 
40  am^y^ 

210  h(?d 
750  ab'c' 

-25affz'' 
- 100  xy ' 


8. 


9. 


10. 


-  7  a'bcd^ 
42  ab^cd'  ' 

St 

2x 
4:X^  —  6  ax 


11. 


Reduce  -— to  its  lowest  terms. 


b' 


Solution. 

§  174,  Prin.  4, 


bx  —  ax  _       x{b  —  g) 
a2-62  ~  {a  +  b){a-b) 

_    —x(a  —  b) 


{a  +  b){a—b)      a  +  b 


X 

a  +  b 


FRACTIONS  131 

Reduce  to  lowest  terms : 

12  ^^-^'  23    ^-^^^-^^ 

•    d'-{-2ab-^b''  '        2x^-2 

-ix+e 


1  o      ^^1 ^^   >   ^  .  24 

4a^-9ar^  05     20-21  a^  +  a;' 
'    8a3  +  27a^'  "   a;^-26a^  +  25' 

J       6xy-3a^y  2^     3  a^H-4  aa;-4  a^ 

*  x^-Sxy'  '   9a^-12ax-h4:x'' 

Sa'b-Sb'  2^    ar^  +  l  +  Sar'  +  Sa; 

*  2b*-2a'b'  .  '     4:-h4:X-x^-Q^ 

17    ^V-f^y±Y.  28.  ^  —  ^^^  —  ^^  +  ^^ . 

^,6  _  ^  ^jj  _   ^^  ^  ^2  -_  ^^ 

18.   A  —  ^f  +  f  gg    am  —  a?t  —  m  +  yt 
a;*  +  2/*  a?n  — a?i4-m  — n 

19     .^^16^1+5^.  3Q    a3-6«-3a^6  +  3a&^ 
'   a^  +  2ic=^-35a;*  *         3a62-3a26- 

^^    7a;-2a^-3  ,,      a:»  +  5  x2-9a;-45 


4-7a;-2a;2  a:3^3^_25a;_75 

2^    ^2^-2^*j-^ ^     l-I/V  ,>  gg  2aa;-av-4  6a;4-2&v 

x^  —  x^ "       *  4:  ax  —  2  ay  —  2  bx  -^  by 

^^     a^-^2a'b  +  ab''  ,^  a^  +  2a^-23  a;-60 


a' -2  a^b^-^ab*  ar^ -  11  a^ - 10 a;  +  200 

185.    To  reduce  a  fraction  to  an  equal  fraction  having  a  given 
denominator  or  a  given  numerator. 

In  order  to  change  f  to  a  fraction  whose  denominator  is  12, 
both  terms  must  be  multiplied  by  12  ^  4,  or  3 ;  similarly,  to 

change  -  to  a  fraction  whose  denominator  is  ns^,  both  terms 

z 

must  be  multiplied  by  nz^-i-z,  or  7)z. 


132  FRACTIONS 


EXERCISES 

186.     1.  Eeduce   ■  ^      to  a  fraction  whose  denominator  is 
a  +  b 

PROCESS 

(a^-b")  -^(a-}-b)  =  a-b. 

Then  g    ^       a(a-b)       ^a'-ab 

'  a-^b      (a-^b)(a-b)      d'-b^' 

Explanation.  —  Since  the  required  denominator  is  (a  —  h)  times  the 
given  denominator,  in  order  that  the  value  of  the  fraction  shall  not  be 
changed  (§  181)  both  terms  of  the  fraction  must  be  multiplied  by  (a  —  &). 

5  a 

2.  Eeduce  —  to  a  fraction  whose  denominator  is  42. 

6 

3  X 

3.  Eeduce to  a  fraction  whose  denominator  is  55  b. 

11  b 

4.  Eeduce  to  a  fraction  whose  denominator  is  84  a;^. 

14  a; 

4  a^ 

5.  E-educe  to  a  fraction  whose  denominator  is  20  'tf. 

5y 

6.  Eeduce     ~'  to  a  fraction  whose  denominator  is  («— 1)1 

a;  — 1 

2x—5 

7.  Eeduce  — to  a  fraction  whose  denominator  is  (2  a;H-5)^. 

2x-{-5 

8.  Eeduce     ^      to  a  fraction  whose  numerator  is  3  a  -j-  a-, 

S  —  a 

9.  Eeduce   ^~-^    to  a  fraction  whose  numerator  is  a;^  —  /. 

2x-i-y 

10.  Eeduce  ^^ —  to  a  fraction  whose  denominator  is  4  —  x^. 

a;  — 2 

11.  Eeduce     ^      to  a  fraction  whose  denominator  is  ?>-  —  9. 

3-5 

12.  Eeduce  a;—  5  to  a  fraction  whose  denominator  is  a;-f-5. 

13.  Eeduce  3  ^+2  Uo  a  fraction  whose  denominatQi:  is  2  Z— .3 1. 


FRACTIONS  133 

187.  Reduction  to  lowest  common  denominator. 

In  algebra,  as  in  arithmetic,  it  is  frequently  desirable  to 
reduce  fractions  that  have  different  denominators  to  respec- 
tively equal  fractions  that  have  a  common  denominator. 

188.  In  algebra,  lowest  common  denominator  corresponds  to 
least  common  denominator  in  arithmetic. 

The  word  'lowest'  has  reference  to  the  degree  of  the  denominator. 

189.  It  is  not  always  easy  to  discover  by  inspection  the 
lowest  common  denominator  (1.  c.  d.),  that  is,  the  lowest  common 
multiple  (1.  c.  m.)  of  the  given  denominators.  However,  it 
may  be  found,  as  in  arithmetic,  by  factoring  the  denominators, 
for  it  is  the  product  of  all  their  different  prime  factors,  each 
factor  used  the  greatest  number  of  times  that  it  occurs  in  any 
denominator.     (See  also  §§  412-414.) 

Thus,  if  the  given  denominators  are  ax  —  hx,  a^— 6*,  and  a*  —  2  a6  f  6^, 
on  factoring  we  find  :  ax—hx  =  x{a  —  h)',  a"^  —  h^  z=  (jot, -\-  6) {a  —  h)\  and 
a2-2a6-H62  =  (a-6)(rt-6). 

Then,  the  factors  of  the  1.  c.  d.  are  or,  a  +  6,  a  —  6,  and  a  —  h. 

Hence,  the  1.  c.  d.  =  a5(a  4-  h)  (a  -  by. 


190.   1.   Keduce  to  fractions   having  their  lowest  common 

denominator:  -  and  -;  -^  and  — ^• 
6         8'  3  6c         6  a6 

PROCESS  PROCESS 

l.c.d.  =  24  l.c.d.=6a6c 

5  =  5.x4^20  j>_^^x2a    ^  2a^ 

6  6x4     24  She     3bcx2a     (5  abc 
3^3x3^^                          _c_^_±xc     ^    c^ 

8     8  X  3     24  6ab      (yab  xc     6abc 

Explanation.  —  The  1.  c.  m.  of  the  given  denominators  is  found  for 
the  1.  c.  d.  in  accordance  with  §  180.  Then,  each  fraction  is  reduced  to 
an  equal  fraction  having  this  denominator,  as  in  §  185. 

Note.  —  All  fractions  should  first  be  reduced  to  lowest  terms. 


134  FRACTIONS 

2.   Reduce  2  m  and  ^"^^  to  fractions  having  their  lowest 
m—  n 

common  denominator. 

Suggestion.  —  First  write  2  m  as  a  fraction  with  the  denominator  1. 

Reduce  to  fractions  having  their  lowest  common  denomi- 
nator : 

8.    J_,  Zl^    J_. 

9    ^  ]jL  ^. 

8  d-c    4  Wc   a?hc^ 


3. 

^"<^¥• 

4. 

%^  and  3  X. 
5  0 

5. 

6.  -5La„d-5_. 
2  icy         4  a?/ 

7.  iMand^S? 

cx  3  62/ 


13.     -.^„,^^, 


10. 

m  —  nf)       a 

?  ^>         ■    • 
a            m-\-n 

11. 

x  +  y    x-y   x'-y' 

2    '     4    '      6 

12. 

X^               X               X 

a^-1'  x  +  l'x-l 

2a 

a* -16'  a2a.4'4_^2 

2a         -2a 


Suggestion.  —  By  §  172,  Prin.  1, 


14. 


15. 


16. 


17. 


4-a^     a2  -  4 
4a       Sb  1 


b—  a   a-\-b   o?  —  b^ 

a  X         —ax 

1  —  ax'  1  +  ax'  ax  —  1 

1  1 


Q^j^l  x  +  l(^'  x^-\-x-2'  x'  +  ^x-b 
3a;  x-1  0^  +  3 


a:2_3^_^2'    x^-^x-^6'    a:^-4x  +  3 


a-f-5  a  — 2  a  +  1 


a2-4a-+-3'  a^-Sa  +  lS'  a^-Ga  +  S 
\ 


FRACTIONS  135 


ADDITION   AND    SUBTRACTION    OF   FRACTIONS 

191.  The  method  of  adding  and  subtracting  fractions  is  the 
same  in  algebra  as  in  arithmetic.  In  algebra,  however,  sub- 
traction of  fractions  practically  reduces  to  addition  of  fractions, 
for  every  fraction  to  be  subtracted  is  really  added  with  its 
sign  changed  (§  64,  Prin.). 

Justas  l  +  ^  =  -i  +  -?-  =  i±^, 

3^4     12^12        12   ' 

a     c      ad  .  be      ad -\- be 

so  —  -|-  —  — —  —  -f-  —  ^ • 

b     d     bd      bd         bd 

1_1^±_  3  ^4-3 

3     4      12      12        12    ' 

a     c  _ad      be  __  ad  — be 

b      d      bd      bd         bd 

EXERCISES 

192.  1.   Add  ^,1^,  and  g. 

Solution.  — Since  the  fractions  have  unlike  denominators,  they  must 
be  reduced  to  fractions  having  a  common  denominator.  By  §  189,  the 
1.  c.  d.  =  60. 

3a;      7x  .  5g_45g      42a;      25g 
4        10       12       60        60        60 


Also  as 


so 


_  46a;  +  42a;  +  2og  _  87x  +  26g 


2.   Subtract  ^::i^  from  5^^+-- 
7  8  4 


Solution 
5x-l      a;      a;- 2  _  35a; -7      14a;      8a;-16 
8  4         7  66  66  66 

^35a;-7  +  14a;-(8a--  16) 

56 
_36a;-7  +  14a;-8a;  +  16_41a;  +  9 
66  66 

Suggestion.  —  When  a  fraction  is  preceded  by  the  sign  — ,  it  is  well 
for  the  beginner  to  inclose  the  numerator  in  a  parenthesis,  if  it  is  a  poly- 
nomial, as  shown  above. 


136  FRACTIONS 

Rule.  —  Reduce  the  fractions  to  similar  fractions  having  their 
lowest  common  denominator. 

Change  the  signs  of  all  the  terms  of  the  numerators  of  fractions 
preceded  by  the  sign  — ,  then  find  the  sum  of  all  the  numerators, 
and  write  it  over  the  common  denominator. 

Reduce  the  resulting  fraction  to  its  lowest  terms,  if  necessary. 

Add :  Subtract : 

_    bm  r         4m 

7.  — —  from 

6  3 

8.  -—  from  — -. 
9  2 

9.  -^from  ?. 
3  y 

10.   ^±±i,om^^. 


3. 

2;  and  ^^. 
5           2    ■ 

4. 

4a      ,  6b 
3   ""S  • 

5. 

2a  ^    ,  3a 
36  ""S6- 

6. 

-'and  -2 
7x           Sx 

Simplify : 


11     2a;  +  l      a;  — 2     a;  — 3  .5-x 


,-     a;-2      x-4:  ,  2-3a;     2a;  +  l 
^^'   ""6  9""^"^  12~' 

a;  — 1      a;  — 2      4a;  — 3  ■  1  —  a; 
^^'  ~3~        18  27     "^     6     * 

2  — 6a;  ,  4a;  — 1      5a;  — 3      1  — a; 

14 . 

5  2  6  3 

,^    a;4-3     x-2  ,  a;-4     a;  +  3 
'^-   ~4  5~  +  l0  6~* 

,^    l-2a  ,  2a-l      2a-a^-\-l 

3  +  a;-a;^     l-a;  +  a;^      l-2a;-2a;^ 

4  6  3 


FRACTIONS  137 

5  a^  4-  52 
18.   Reduce     „     ,„ —  2  to  a  fraction. 
a^—b^ 

Solution 

6  gg  4.  ?)2      2  ^  5  g^  +  &^  -  2(q2  -  &2) 
a:i--b-^       1  g-^-6-2 

_  5  gg  +  ft2  _  2  g2  +  2  &2 

g2  -  6-^ 
Reduce  the  following  mixed  expressions  to  fractions : 


19. 

a 

-I- 

20. 

X- 

-V. 
2 

21. 

<l 

-c" 

4- 5  c. 

23. 

a      «^'-«^ 
0 

24. 

„_.r:|^. 

25. 

an- a; 

a  —  a; 

26. 

a^-ab-^b^-- 

b' 
1 

c 

22.   lll^-4a;.  

3  a  +  6 

Perform  the  additions  and  subtractions  indicated  : 

-    »-f>  .b-c  33    l^i^^^_2. 

X  l-\-x 

Sa'-4a^ 


ab          be 

28. 

a-\-b      a  —  b 
a—b      a+6 

29. 

x-y 

30. 

x         x-2 

x-2      x-\-2 

31. 

x—1 

32. 

m f-n, 

34.    3  a  -  2 


a; 


35. 


3a-\-2x 
1  1         2 


a:  —  1      a;  -f  1      x- 


36.    -t i-+     2a 


a  +  6      a  — 6     a'^  —  b- 


o  —  a;      a-{-x      a^  —  xr 


m  —  n  ax 


38.   3.'B  +  -^-(^2a;4-— \ 
V    -      <^7 


138  FRACTIONS 


39.    -^ ^^^=^4-      ^ 


2     a  +  2     4-a2 
Suggestion.  —  By  §  172,  Prin.  1,  — — 


40.   ^±1+-^+    4a 


a— 1      a+1      1 


41.  i::U[L^  + 

a;2_4  ^ x-2     2-x 

42    ^(a+^_3a^^-^^4^^ 


43. 


a  —  x  X—  a 

a  a  2ab         4a6^ 


a  —  b      a-\-b      a^  +  b^     a*-\-b* 


Suggestion.  —  Combine  the  first  two  fractions,  then  the  result  and  the 
third  fraction,  then  this  result  and  the  fourth  fraction. 

..    a-{-b     a  —  b       4  a6     ,    Sab^ 

44.    — ■ -f- 


a-b     a  +  6     a'  +  b^     a'-\-b* 

^^        1             1  26      ,     2b' 

45. — -    o  ■   ,o  + 


a-b     a  +  6     a'  +  b'     a*-\-b* 

4g  a^  +  y y  +  g  J  g-ha? 

(y-z)(z-x)      (x-z)(x-y)      {y-x){z-y) 

Solution 

Sum= '^±y + y  +  ^         + ?-±^ 

{y-z){z-x)      (z-x){x-y)      {x-y){y'-z) 

= ^- =  0. 

ix-y)iy  -z)iz-^) 


(6-c)(a-c)      (c-a)(a-6)      (6-a)(6-c) 
48.    ^+i + ^-±1 +         "  +  1  - 


(a  —  b){a  —  c)      (6  —  c)  (6  —  a)      (a  —  c)(b  —  c) 


FRACTIONS  189 


MULTIPLICATION   OF   FRACTIONS 

193.  Fractions  are  multiplied  in  algebra  just  as  they  are  in 
arithmetic. 

Thus,  7X0= :; — n ' 

'  4     2     4x2 

In  general,  1X3  =  7^-     That  is, 

0     a     oa 

Principle.  —  The  product  of  two  or  more  fractions  is  equal  to 
the  product  of  their  numerators  divided  by  the  product  of  their 
denominators. 

EXERCISES 
X 


194.   1.   Multiply  -^-^  by  a^-25. 
x-\-o 

Solution 


r2^-^  =  (x-6)2  =  a;2-10a;+25. 


2.   Multiply  ?^  by  1 


+ 


x-\-2  x-\-l 

Solution 

\x  +  2)\       x  +  \)     x  +  2\x  +  l      x  +  lj 
^x  +  Z    Z"^ 

jM^.  x  +  1 

x  +  l' 

General  Suggestions.  —  1.  Any  integer  may  be  written  with  the 
denominator  1. 

2.  After  finding  the  product  of  the  numerators  and  the  product  of  the 
denominators  the  resulting  fraction  may  be  reduced  to  lowest  terms,  in 
many  cases,  by  canceling  common  factors  from  numerator  and  denomi- 
nator. It  is,  however,  more  convenient  to  remove  the  common  factors 
before  performing  the  multiplications. 

3.  Generally,  mixed  numbers  should  be  reduced  to  fractions. 


140 


FRACTIONS 


Multiply : 

3. 

i  ^^  I 

4. 

>^h 

■ 

S. 

2y 
3  a' 

6. 

^^■1  by 
2ae    ^ 

Sax 

7. 

10  c^  ^ 

3  be 
a' 

8. 


9. 


10. 


11. 


12. 


4mr? 
2  ax 


bv 


15  bx 

16  m2 


12  by    ^         x" 


a 


by 


a  +  6 

x]f- 
20-8« 


a  — 6 

25  -  10a; 


by 


a^y 


l-6a;  +  5a.'^  b     ^~^ 
a^-3a;  +  2      "^  l-a;* 


Simplify  each  of  the  following : 


\ 


13. 


15. 


18. 


20. 


21. 


a  +  5 

,4         ^4 


-  a6      a'  -  M 


14.  ^;;i£:x4±^„x^ 


aa;  +  ^ 


a^  +  ar^ 
4a 


2x  +  y     Aa^  —  ab 


2a 


(a  +  xy 
4:X^  —  y^ 


16    i>  +  2     3a^-27  4 

•    a._3       2jp2_8       ^a;-f3p 


17.  ^llZ^v-^^^X 


7>^ 


{p-qf      lf~\-pq      f  +  (f 

a^  +  8      a^  +  2a  +  4 
a3_8     a2_2a  +  4* 


19.   ^^l±_^^!^!dt^  X 


a*  —  aa^ 

a^  +  4 


a^  —  ax  -f  a^ 

a^  +  a  +  l 


a^  +  a^  +  l      a2  +  2a  +  2 


4^2-4 


lOr  +  lOa 

5r2  +  10rs  +  5s2^  Sr^-Ss^ 


FRACTIONS  141 


22. 


23.    fl  + 


7  a;  + 11     y.  _     17  a; -11    \ 


24.    - 


25. 


26. 


27. 


4  aar^  —  4  a?/^ ^  3  aa^  —  3  oary  +  bxy  —  by^ 

3  a^  +  3  axy  -j-  6a;?/  -f  by-       5  aar^  — 10  axy  +  5  ai/^ 

ar3_5a;2_^8a.'-4      a^-lQ  3^4-33  a;-3G 
a^-Sx'-{-19x-12'    x'-6x^-\-nx-6   ' 

x*-Sa^-2Sx^  +  75x-50    ^  ar^  -  10  x*  +  29  a;  -  20 
a.*_5ar»-21a;2  +  125a;-100  *  ar^ - 12 ar^  +  45  a; - 50  ' 

a^ -\-ab  +  ac-\-bc  ^  a^  —  ax 4-  ay  —  xy  ^  a^—a;( ?/  —  «)—  ay 
ax  —  ay  —  x-+xy     d^ -\- ac  +  ax -^  ex     a^  —  a{y  —  b)—by 

DIVISION   OF   FRACTIONS 

195.  The  reciprocal  of  a  fraction  is  the  fraction  inverted,  or 
1  divided  by  the  fraction. 

2        ^         X       z  ni         \ 

The  reciprocal  of  -  is  - ;  of  -  is  - ;  of  m,  or  — ,  is  —  • 
S       2         z       X  1         m 

196.  The  reciprocal  of  a  number  is  1  divided  by  the  number. 

m.  Justas  M  =  fx|  =  |^^, 

a     c      a.d     ad 
so  --j--  =  -X-  =  —  • 

b      d      b      c      be 

Principle.  —  Dividing  by  a  fraction  is  equivalent  to  multiply- 
ing by  its  reciprocal. 

EXERCISES 

198.    Write  the  reciprocal  of  : 


a 
b' 

3.   rs. 

5.   Ztu. 

7. 

4 

ab' 

.     3m 
P 

be 

6.       1    . 

3m 

8. 

a  —  x 
h-y 

142  FRACTIONS 


9.   Dmde-^by_^ 


Solution 

a;2  -  4     a;  +  2  ^  Ca>K2r)(a;  -  2) 


x'^  -I     x  —  1     (x  +  1)  (a>-^T;    >-r2    X  +  1 

Simplify : 

10     5  ^^  .  ^Q  ^^^  ^K  g*  — 6^         .    a^  +  6^ 

'     6bx   '    3  ax''  '    a^-2ab  +  b^  '  a'-ab' 

11.    ^-^^-^abx.  16.     ^±£^^!±a±l!.      i^ 

7  x'  —  y'  X—  y 

(m  +  z/)2     m^-?/2  \       yj     \        ^y 

a  +  b  b  \b         J     \b^         J 

Suggestion.  —  Reduce  the  dividend  to  a  fraction. 


24. 


.2  _  />2 


ar^  +  2ar^-19a;-20  '  a^  +  lO  a;^  ^29a;  +  20' 


FRACTIONS  143 


Complex  Fractions 

199.    A  fraction  one  or  both  of  whose  terms  contains  a  frac- 
tion is  called  a  complex  fraction. 


EXERCISES 


200.   1.  Simplify  the  expression 

y 

a 

-  b      a  .  X     a  ,v     ay 

Solution.  — =--^-=-x^  =  -*. 

£:       0     y      b     X     bx 

y 

Simplify : 
2     -^. 


3. 


t+« 


8.    Simplify  the  expression  ^ — 

^  +  -  +  1 

r    2/ 


Solution.  —  On  multiplying  the  numerator  and  denominator  of  the 
fraction  by  y'^,  which  is  the  1.  c.  d.  of  the  fractional  parts  of  the  numera- 
tor and  denominator,  the  expression  becomes  ^  —xy  +  y' 
#x.  a;2  +  xy  +  y2 


^-      -> 


144  FRACTIONS 

Simplify: 

x^  —  1  jc^  +  y^                      x^  +  y^ 

X  xu                              2y 

a; 4-1  ar  —  .r//  +  y^                       ^  _^ 

a."^  icy                               y     X 

1 1  1  1                   1 


10.    T-^T—  12.       ""  +  :      •  14.    -i— ^. 

-  — — —  -J  1  g 

^     2/  +  2  a  +  1  .      a-1 

1  1  1 

ic+l       ,  a;-hl      1-a; 
15. 1 1 


1  +  a;      1  — a;      1  +  a; 


^4-^ 


16.  ?_A+£^  1 


^^b'-^(f-a' 


17.    Simplify  the  expression 


a      b-\-c  2  be 

1 


1+-^ 


X 


Solution. — By  successive  reductions  and  divisions, 

1  1  1  x+1  x+1 


1+_J_     i+-JL_      1  |.     ^        x+li-x     2a;+l 

1  +  1  £±I  ^  +  1 

X  a; 

Simplify :       ' 

1  2 

18. 20.    


x-\- 


2 


l+-^  +  l 


3-a?  2-x 


1  x-2 

19.   7-  21. 


a-\ ^-  a;  — 2 


.  1  a;-l 

a  a;  —  .6 


FRACTIONS  145 


EQUATIONS   AND    PROBLEMS 

201.  Since  the  student  has  learned  how  to  perform  opera- 
tions when  fractions  are  involved,  he  is  now  prepared  to  solve 
certain  equations  that  heretofore  he  could  solve  only  by  a 
roundabout  method,  and  others  that  he  could  not  solve  at  all. 

Clearing  Equations  of  Fractions 

202.  The  process  of  changing  an  equation  containing  frac- 
tions to  an  equation  without  fractions  is  called  clearing  the 
equation  of  fractions. 


203.    1.    Solve  the  equation  ^  =  10- 1 


Solution 


?=10_?. 
2  3 

Since  the  first  fraction  will  become  an  integer  if  the  members  of  the 
equation  are  multiplied  by  2  or  some  number  of  times  2,  and  since  the 
second  fraction  will  become  an  integer  if  the  members  are  multiplied  by 
3  or  some  multiple  of  3,  the  equation  may  be  cleared  of  fractions  in  a 
single  operation  by  multiplying  both  members  by  some  common  multiple 
of  2  and  3,  as  6,  or  12,  or  18,  etc. 

It  is  best  to  multiply  by  the  1.  c.  m.  of  the  denominators,  that  is,  by 
the  Led.  of  the  fractions,  which  in  this  case  is  6. 

Multiplying  by  0,  Ax.  3,  3  a;  =  60  -  2  x. 

Transposing,  etc.,  §  71,  5  a:  =  60. 

Hence,  Ax.  4,  a;  =  12. 

VerificatioiJ.  —  When  12  is  substituted  for  x,  the  given  equation  be- 
comes Q  =  (S  \  that  is,  the  equation  is  satisfied  for  x  =  12. 

«cii      i-i,  i.-       x  —  1      x  —  2     2     x  —  3 

2.   Solve  the  equation  — — -  = 

Z  o  o  4 

Suggestion.  —  Multiplying  both  members  of  the  equation  by  the 
I.  c.  d.,  which  in  this  case  is  12,  we  obtain 

6(.r  -  1)- 4(x  -  2)  =  8  -  3(a;  -  3). 

MILNE's    IST    YR.    ALG.  —  10 


146  FRACTIONS 

To  clear  an  equation  of  fractions : 

Rule.  —  Multiply  both  members  of  the  eqimtion  by  the  lowest 
common  denominator  of  the  fractions. 

Cautions.  —  1.  To  insure  correct  results  in  solving  equations  : 

Before  clearing,  reduce  all  fractions  to  lowest  terms,  and  unite  frac- 
tions that  have  like  denominators. 

Test  results  and  reject  such  as  do  not  satisfy  the  equation. 

2.  If  a  fraction  is  negative,  the  sign  of  each  term  of  the  numerator 
must  be  changed  when  the  denominator  is  removed. 

Solve,  and  verify  each  result : 


-  ^^+l=f  •        -  l^l=f 

4.  ?+io  =  i3.              ■      6.  ri-f|=f 

7. 

XX     a;     3a;      5  a;      „ 
2     3     4      10      12~ 

8. 

25  a;     5  a;     2  a;     ^^_2 
18        9        3        6 

9. 

72!  +  2     12-2     2  +  2      g 
6              4*2 

10. 

w-3     it  +  5     w  +  2      . 
7*3            6 

11. 

y-1     y-2     2/-3_52/-l. 

2      '      3      '      4              6 

12. 

a;  — 5      2a;  +  2      a;— l_a;  +  4 

3  8  4  6 

13.  1.07  a; +.32  =.15  a; +  10.12 +  .675  a;. 
Suggestion.  —  Clear  of  decimal  fractions  by  multiplying  by  1000. 

14.  .604 a; -3.16 -.7854 a; +  7.695  =  0. 
.2a;     .la;     .la;  .  .4a;_  .3 

*  "T    X    "2~'^T~~14' 


FRACTIONS  147 


14         6  a; +  2  d6  14 

Suggestion. — The  equation  may  be  written, 

^      '  14      14      6  X  +  2       56       56      56 '       Qx  +  2      14 

,^    3a;-2  ,  3a;-21      6a;-22 


18. 


19. 


2a; -5  5  10 

4a;  +  3^8a;4-19     7a;-29 
9  18  5a;-12 

6p^-{-p      2p-4  ^2j>-l 
Wp        7p-13  5 


20.  Solve  the  equation  ^^  +  ^^^  =  ^^  +  ^— ^  • 
x  —  2     x  —  7     x—i)     x  —  3 

Solution.  —  It  will  be  observed  that  if  the  fractions  in  each  member 
were  connected  by  the  sign  — ,  and  if  the  terms  of  each  member  were 
united,  the  numerators  of  the  resulting  fractions  would  be  simple.  The 
fractions  can  be  made  to  meet  this  condition  by  transposing  one  fraction 
in  each  member  before  clearing  of  fractions. 

x-1      x—2      x-^     x—6 


Transposing, 
Uniting  terms. 


2      x-S      x-Q      X 
-1  -1 


a;2-5a;-f-6      x^-lSx  +  42 

Since  the  fractions  are  equal  and  their  numerators  are  equal,  their 
denominators  must  be  equal. 

Then,  a;2  -  5  a;  +  6  =  a;2  -  13  a;  +  42. 

.-.  x  =  il 


21. 


22. 


23. 


X 

-1 

X 

-7 

X  — 

5 

X 

-3 

X 

-2 

+^ 

-8" 

X  — 

^ 

+x 

-4 

X 

-S 

X 

-7 

X  — 

8 

X 

-4 

X 

-4 

+  x 

-8" 

X- 

^ 

+^ 

-5 

V 

^2 

V' 

+  3 

v  + 

5 

V- 

4-6 

v  +  l     v  +  2     v  +  4     v-h5 


148  FRACTIONS 

Algebraic  Representation 

204.    1.   What  part  of  m  —  n  is  j9  ? 

2.  Indicate  the  sum  of  I  and  m  divided  by  2,  and  that  result 
multiplied  by  n. 

3.  Indicate  the  product  of  s  and  (r  —  1)  divided  by  the  nth 
power  of  the  sum  of  t  and  v. 

4.  A  boy  who  had  m  marbles  lost  -  of  them.  How  many 
marbles  had  he  left  ? 

5.  By  what  number  must  x  be  multiplied  that  the  product 
shall  be  2  ? 

6.  Indicate  the  result  when  the  sum  of  a,  6,  and  —  c  is 
to  be  divided  by  the  square  of  the  sum  of  a  and  b. 

7.  It  is  t  miles  from  Albany  to  Utica.  The  Empire  State 
Express  runs  s  miles  an  hour.  How  long  does  it  take  this 
train  to  go  from  Albany  to  Utica  ? 

8.  A  cabinetmaker  worked  x  days  on  two  pieces  of  work. 
For  one  he  received  v  dollars,  and  for  the  other  w  dollars. 
What  were  his  average  earnings  per  day  for  that  time  ? 

9.  A  train  runs  x  miles  an  hour  and  an  automobile  x—  y 
miles  an  hour.  How  much  longer  will  it  take  the  automobile 
to  run  s  miles  than  the  train  ? 

10.  Indicate  the  result  when  h  is  added  to  the  numerator 

and  subtracted  from  the  denominator  of  the  fraction  -  • 

c 

11.  A  farmer  had  -  of  his  crop  in  one  field,  -  in  a  second, 
^  X  y 

and  -  in  a  third.     What  part  of  his  crop  had  he  in  these  three 
z 

fields  ? 

12.  A  student  spends  —  of  his  income  for  room  rent,  —  for 

m  n 

board,  -  for  books,  and  -   for  clothing.     If  his  income  is  x 
s  r 

dollars,  how  much  has  he  left  ? 


FRACTIONS  149 

Problems 
205-    Solve  the  following  problems  and  verify  each  solution : 

1.  If  a  large  lemon  grown  in  Mexico  had  weighed  2^  pounds 
less,  its  weight  would  have  been  |  of  its  actual  weight.  What 
was  its  actual  weight  ? 

2.  The  crew  of  the  Lusitania  numbers  800.  If  this  is  200 
less  than  \  the  number  of  passengers  and  crew  that  may  be 
accommodated,  what  is  the  passenger  capacity  ? 

3.  The  box  of  a  Chinese  sedan  chair  is  1^  feet  higher  than 
it  is  long  and  the  area  of  ^fclrc  floor  is  4  square  feet.  Find  its 
three  dimensions,  if  the  capacity  of  the  box  is  14  cubic  feet. 

4.  The  sum  of  the  heaviest  loads  that  can  be  carried  by  a 
man,  a  horse,  and  an  elephant  is  2900  pounds.  The  elephant 
can  carry  10  times  as  much  as  the  horse,  and  the  horse  If 
times  as  much  as  the  man.     What  load  can  each  carry  ? 

5.  The  first  issue  of  Tlie  Sun  devoted  \  of  its  columns  to 
advertisements  and  \  to  miscellaneous  news.  The  rest  of  the 
paper,  5  columns,  was  devoted  to  poetry,  finance,  and  shipping 
news.     How  many  cohimns  did  it  contain  ?  1  "^^ 

6.  Of  the  world's  supply  of  rubber  one  year.  South  America 
produced  \  and  Africa  \.  How  much  was  produced  by  each, 
if  the  rest  of  the  world  produced  26,600  tons  ? 

7.  A  large  ear  of  seed  corn  exhibited  at  the  Iowa  Experiment 
Station  sold  for  ^150.  At  the  same  rate,  if  it  had  weighed 
8  ounces  less,  it  would  have  sold  for  $  90.  How  much  did  it 
weigh  ? 

8.  The  feathers  that  a  Toulouse  goose  yields  in  a  year  are 
valued  at  $2.80.  If  it  yielded  4  ounces  more,  they  would 
be  worth  f  3.50.     Find  the  weight  of  the  yield  of  feathers. 

9.  The  number  of  vessels  entering  at  New  York  in  one  year 
was  11,399.  If  \  of  the  number  of  steamships  was  449  more 
than  \  of  the  number  of  sailing  vessels,  how  many  of  each 
were  there  ?  '    i 


150  FRACTIONS 

10.  Find  the  world's  production  of  nickel  in  a  year  when 
the  United  States  and  Canada  together  produced  ^  of  it,  Eng- 
land ^  of  it,  and  the  rest  of  the  world  3800  tons. 

11.  In  a  recent  year,  the  United  States  produced  2  times  as 
much  aluminium  as  Germany,  1^  times  as  much  as  France,  and 
6200  tons  more  than  England.  If  all  these  countries  produced 
19,800  tons,  how  much  did  the  United  States  produce  ? 

12.  The  largest  log  in  a  shipment  of  mahogany  sent  to  New 
Orleans  weighed  14,000  pounds.  If  the  weight  of  the  rest 
of  the  shipment  had  been  1  ton  more,  the  weight  of  this  log 
would  have  been  ^^  of  the  total  weight  of  the  shipment.  Find 
the  weight  of  the  shipment. 

13.  In  constructing  the  Hall  of  Records  building  in  New 
York  City,  600,000  pounds  of  copper  were  used.  The  dome 
lacks  5250  pounds  of  having  ^  as  much  as  the  rest  of  the  build- 
ing.    How  many  pounds  of  copper  are  there  in  the  dome  ? 

14.  The  freight  charges  on  a  car  load  of  hay  were  ^  as 
much  as  on  a  car  load  of  apples.  If  there  were  10  tons  of 
hay  and  the  charges  on  each  ton  were  f  2  less  than  ^  of  the 
charges  on  all  the  apples,  find  the  charges  on  each  car  load. 

15.  The  number  of  pound  cans  of  salmon  in  a  case  is  4  more 
than  Jg  of  the  number  of  cans  that  can  be  packed  in  a  minute 
in  a  Washington  cannery.  If  1000  cases  can  be  packed  in  an 
hour,  how  many  cans  are  there  in  a  case  ? 

16.  A  boy  sold  from  his  garden  a  certain  number  of  bunches 
of  beets.  If  he  had  sold  7  bunches  more  he  would  have  re- 
ceived $  11  for  them.  If  he  had  sold  5  bunches  less  he  would 
have  received  $  9.80.  How  many  bunches  did  he  sell  and  at 
what  price  ? 

17.  if  the  number  of  pounds  of  alligator  teeth  sold  in  a 
given  year  had  been  50  less,  the  approximate  number  of  teeth 
would  have  been  14,000 ;  if  200  less,  the  number  of  teeth  would 
have  been  3500.  Find  the  number  of  pounds  sold  and  the 
average  number  of  teeth  in  a  pound. 


REVIEW  151 


REVIEW 


206.    1.    What  three  signs  are  to  be  considered  in  connection 
with  a  fraction  ?     What  is  the  sign  of  a  fraction  ? 

2.  Under  what  conditions  may  the  sign  of  the  numerator 
or  of  the  denominator  of  a  fraction  be  changed  ? 

3.  Show  that     ^  ~^ 


4.  What  is  the  effect  of  changing  the  sign  of  an  odd  num- 
ber of  factors  in  either  term  of  a  fraction  ?  an  even  number  ? 

5.  Show  that  (^-y)(^-y)  =  -^^"ixy-f^ 

(z  —  w)(u  —  z)        {z  —  w)(z  —  u) 

6.  When  is  a  fraction  in  its  lowest  terms  ?  What  principle 
applies  to  the  reduction  of  fractions  to  higher  or  lower  terms  ? 

7.  Reduce  to  lowest  terms : 

4aV  +  12a6ar^-f9  6V  36a^-1962^ 

8.  In  reducing  a  fraction  to  an  equal  fraction  having  a 
given  denominator,  how  is  the  number  found  by  which  both 
terms  are  to  be  multiplied  ? 

Q    7, 

9.  Reduce    to  a  fraction  whose  denominator  is 

a-2b 
a2_4a6+462. 

10.  Define  highest  common  factor ;  lowest  common  multiple. 
Illustrate  by  finding  the  highest  common  factor  and  the  low- 
est common  multiple  of  ax  -\-ay,  a^  —  y^,  and  a^ -\- 2  xy -{- y^. 

11.  What  is  the  reciprocal  of  a  fraction  ?    of  any  number  ? 

Give  the  reciprocals  of  -,  ^"*"    ,  and  x, 
y   x-y 

12.  Define  complex  fraction  and  illustrate  by  writing  one. 
Simplify  the  one  you  have  written. 


152  REVIEW 

Reduce  to  an  integral  or  a  mixed  expression: 
13     a^'-9/H-7  ^^     4a'+20a'b-\-27  ab'+9b^ 

x-3y      '  '  2a+36 

Reduce  the  following  mixed  expressions  to  fractions : 

15.   x'-xy  +  f---^^.  16.    !^^1±^' -  ,1  -  m. 

x-^y  m-\-7i    ■ 

Simplify : 

17.  iiJ  +  'lui.    ^+^' 


2rl    '    4.rl      Arl{s-{-ty 

18     <^  +  ^  .(^  —  x      2(x~  —  2  a) 
xT2      2^^  x^-4: 

19.  (^'-f^^^  +  ^y\.f    ^''y  +  ^y^    x^'^'^\ 

\xy  +  y'       x-y  J  '  \x^  +  2 xy '-\- y^         y'    J 

•    [a'-2ab-\-b'     a-bj^\     2b        '2a-2bj 

21.  What  is  meant  by  clearing  an  equation  of  fractions  ? 
State  the  axiom  upon  which  it  is  based. 

22.  What  precautions  must  be  taken  to  secure  correct  re- 
sults in  solving  equations  that  involve  fractions  ? 

Solve,  and  verify  each  result: 

23.  ^-1^  =  6.  25.     £±J  =  Bi^  +  l. 

2        4  ^2 

24.  20a:  +  i^  =  ^.  26.    ^-^^  1 1:!?^*  =  ^11^. 

3  6  10  o  15 

x^-3      X     12      x-S 


27. 


4      24  4 


28.  ^±I^^±_§^3a:-1.5_ 
a;  +  2^     3  9 

29.  •  2.04  a;  -  3.1  -  2.95  x  =  8.12  -  5  a;  + 1.05. 


SIMPLE   EQUATIONS 


ONE   UNKNOWN   NUMBER 

207.  The  student  already  knows  what  an  equation  is;  he 
has  solved  several  different  kinds ;  and  he  knows  some  of  the 
kinds  by  name.  In  this  chapter  and  the  next  he  will  meet 
some  of  the  same  kinds  with  the  treatment  extended  to  a  few 
new  forms  and  some  additional  methods  of  solution. 

208.  An  equation  that  does  not  involve  an  unknown  num- 
ber in  any  denominator  is  called  an  integral  equation. 

X  +  5  =  8  and f-  6  =  8  are  integral  equations.     Though  the  second 

o 

equation  contains  a  fraction,  the  unknown  number  x  does  not  appear  in 

the  denominator. 

209.  An  equation  that  involves  an  unknown  number  in  any 
denominator  is  called  a  fractional  equation. 

8             2x 
x  +  6  =  -  and  =  7  are  fractional  equations. 

X  X—  1 

210.  Any  number  that  satisfies  an  equation  is  called  a  root 
of  the  equation. 

2  is  a  root  of  the  equation  3  x  +  4  =  10. 

211.  Finding  the  roots  of  an  equation  is  called  solving  the 
equation. 

212.  Two  equations  that  have  the  same  roots,  each  equation 
having  all  the  roots  of  the  other,  are  called  equivalent  equations. 

X  -f  3  =  7  and  2  x  =  8  are  equivalent  equations,  each  being  satisfied  for 
X  =  4  and  for  no  other  value  of  x. 

163 


154  SIMPLE   EQUATIONS 

Numerical  Equations 

213.  By  applying  axioms  to  the  solution  of  equations,  the 
endeavor  is  made  to  change  to  equivalent  equations,  each  sim- 
pler than  the  preceding,  until  an  equation  is  obtained  having 
the  unknown  number  in  one  member  and  the  known  numbers 
in  the  other. 

Solve,  and  verify  each  result : 

1.  8^  =  24.  5.    11+0^  =  15.  9.   4/1  +  3  =  7. 

2.  9r=54.  6.    20+a;  =  30.  10.    6  r  -  7  =  5. 

3.  ^r  =  1.5.  7.    72/-5  =  2.  11.   16  +  3  =  8. 

4.  ia;  =  2.5.  8.    22  +  3  =  9.  12.   ia;  +  2  =  6. 
13.    8aj-7  =  3  +  6a;.                    18.    17  ^  +  5(2  -  3  i)  =  18. 


14.  7a;  +  6  =  6a;  +  8.  19.   5  a;- (4  -  6  a;- 3)  =26. 

15.  5a;-10  =  2a;  +  20.  20.    (2w-l)=^  =  4(^-3)1 

16.  4r-18  =  20  +  |r.  21.    21x-\-{x-4.f==(p  +  x)\ 

17.  5n-(2n  +  3)  =  12.  22.    (12  a;  +  6)  H-3  =  9-3a;. 

23.  (aj  +  l)(a;  +  2)  =  ll  +  .'«2. 

24.  \x-4.  +  \x  =  lQ  +  \x-10. 

25.  fl;(a5  +  5)-6  =  a;(a;-l)  +  12. 

26.  3(2-a;)-2(a;  +  3)=6-2a;. 

27.  aj-(2  +  4a;)  =  13-5(aj  +  5). 


28.    2Sa;-2a;-2S=3ja;-(3x-3)J. 

29.  6a;-13-9a5  +  a;  =  4x-12  +  3a:-6a;-13. 

30.  36  +5  aj-  22  -  (7a;-  16)  =  5  a;  +  17  -  (2  a;  +  22). 

31.  2(r-5)(r-4)=(r-4)(r-3)  +  (r-2)(r-5). 

32.  12a;-(6a;-17a;-15-a;)  =  15-(2-17a;+6aj-4-12a;). 


SIMPLE   EQUATIONS  165 


33.    3.T-?  =  14.  34     ^-f  =  f 

o  3       6     4 

2x  7 X     5x      X  _4 

'     3  8       18      24~9' 


36. 


3^-5     "^t-lS^o     t-\-3 
4  6  2 


37.    r(2_r)-^(3-2r)  =  ^'^^^ 


38. 


2^         '      4^  '6 

6r  +  3       3r-l  ^2r-9 
15         5r-25  5 


39     ^  +  1   ,  g  +  6^s  +  2      g  +  5^ 
s  +  2      s-f7      .s-i-3      s  +  6* 


Literal  Equations 

214.    1.   Solve  the  equation  ^11:^  =  ^11^  for  x. 

ah 

Solution 

a     ~      h     ' 

Clearing  of  fractions,  hx—h^  =  ax  —  a^. 

Transposing,  etc. ,  ax—hx  =  a^  —  6«. 

(a-6)x  =  a3-63. 

Dividing  by  (a  -  6) ,  x  =  a^  +  ab  +  62. 

Verification.  —  Since  a  and  b  may  have  any  numerical  value,  let 
a  =  2  and  6  =  1;    then  x  =  a'^  +  ab-\-b^  =  i  +  2  +  l  =  7,  and  the  given 

equation  becomes  ■  ~     =  '  ~    ,  or  3  =  3;  consequently,  the  equation  is 
satisfied  for  a;  =  a^  +  a6  +  bK 


156  SIMPLE   EQUATIONS 

Solve  for  a;,  and  verify  each  result : 

_     c^—x.n-      1                         ^     X      x  +  2h      a      o 
z. 1 — _-.  a. =  -  — c). 

nx        ex     c     ,  hah 

_-,       ah  _1       49                   ^     x-a  ,  2x     r  ^  ^^ 
6.    1 —  •  y.    ■ — 1 =  o-\ 

X-     ah     ahx  ha  a 

4.  rx-\-s'  =  9^  -sx.     .  10.    6-\-l-2x  =  l(x-2). 

5.  a^x-h'  =  h^x-a\  11.    c'x  +  d'' =  c^' -^  d'x. 

a^-\-h^     a  —  h__h  -„     x  —  2  a     x  _a--^h^ 

2hx        2  hx^      X  a  h         ah 

^     2x-a     x-o-^^  ^^     a'       h^  ^a  +  h     3(a4-6) 

X—  a      x  +  a       '  '    hx     ax       ah  x 

14.  6x  +  lS(l  —  ^a)  =  a{x  —  a). 

15.  a:\a  —  x)  =  ahx  +  h\h-\-  x). 

16.  h(2x-9c-14:h)  =  c(c-x). 

17.  a(x-a-2h)-^h(x-h)-^c{x  +  c)  =  0. 

18.  (a  —  x)(x—'b)-\-(a-\-x)(x  —  h)  =  (a  —  hy. 

19.  (?/i  H-  a;)^  +  (7>i  -f-  a;)(?i  —  ic)  =  (m  +  ?i)^ 

20.  (a  —  h)(x  —  c)  —  (h  —  c)(x  —  a)  =  {c  —  a)(x  —  h). 
a  —  h-{-c     h  —  a-{-c 


21. 


22. 


23. 


24. 


x—1      a—1  a?  —  a^ 


a  —  1     x  —  1      {a  —  l){x—l) 

1  2  mr^  m  x  —  n 


a-\-x       2x        x^(x  —  a)_t 
a         a-\-x     a(a^  —  x^     3 


Suggestion.  —  Simplify  as  much  as  possible  before  clearing  the  equa- 
tion of  fractions. 


25       ^(^-^)    ,^Q>-^)_^-^  ^Q 
•     12Q,2_^^-^  h{x-b)  h' 


SIMPLE   EQUATIONS  -         157 

Problems 

215.  Review  the  general  directions  for  solving  problems 
given  on  page  45. 

1.  What  is  the  weight  of  a  turtle,  from  which  6}  pounds  of 
tor^ise  shell  is  taken,  if  this  is  ^^^  of  the  turtle's  whole  weight  ? 

2.  The  powder  and  the  shell  used  in  a  twelve-inch  gun 
weigh  1265  pounds.  The  powder  weighs  15  pounds  more  than 
J  as  much  as  the  shell.     Find  the  weight  of  each. 

3.  One  day  three  lace  makers  earned  80  cents.  The  be- 
ginner earned  \  as  much  as  the  expert  maker,  and  the  average 
worker  earned  3  times  as  much  as  the  beginner.  How  much 
did  each  earn  ? 

4.  One  ton  of  coal  will  make  8.7  tons  of  steam.  If  the 
Lusitania  requires  1200  tons  of  coal  a  day  for  this  purpose, 
how  many  tons  of  steam  are  required  an  hour  ? 

5.  A  grocer  paid  $8.50  for  a  molasses  pump  and  5  feet  of 
tubing.  He  paid  12  times  as  much  for  the  pump  as  for  each 
foot  of  tubing.     How  much  did  the  pump  cost?  the  tubing? 

6.  In  lighting  a  hall  a  certain  number  of  16-candle  power 
electric  lamps  and  twice  as  many  20-candle  power  lamps  were 
used.  The  total  illumination  amounted  to  224  candle  power. 
Find  the  number  of  lamps  of  each  kind  used. 

7.  At  the  waterworks  2  large  pumps  and  4  small  ones  de- 
livered 4800  gallons  of  water  per  minute.  Each  of  tlie  large 
pumps  delivered  4  times  as  much  water  as  each  small  pump. 
How  many  gallons  per  minute  did  each  pump  deliver  ? 

8.  The  crew  of  a  United  States  battleship  in  target  practice 
made  11  hits  in  less  than  a  minute.  If  J  of  the  number  of 
shots  fired  was  9  times  the  number  of  misses,  how  many  shots 
were  fired  ? 

9.  The  courtyard  of  a  palace  is  101  feet  longer  than  it  is 
wide.  If  its  width  were  decreased  25  feet,  its  length  would  be 
twice  its  width.     Find  the  dimensions  of  the  courtyard. 


158         -  SIMPLE   EQUATIONS 

10.  In  making  5000  pounds  of  brass  there  were  used  81 
times  as  much  copper  as  tin,  and  twice  as  much  tin  as  zinc. 
How  many  pounds  of  each  metal  were  used  ? 

11.  A  merchant  bought  62  barrels  of  flour,  part  at  $4f  per 
barrel,  the  rest  at  $5^  per  barrel.  If  he  paid  $320  for  the 
flour,  how  many  barrels  of  each  grade  did  he  buy  ? 

12.  A  dealer  paid  $  185  for  25  boxes  of  candles.  If  he 
paid  $  9  a  box  for  part  of  them  and  $  6.50  a  box  for  the  rest, 
how  many  did  he  buy  at  each  price  ? 

13.  A  merchant  purchased  an  assortment  of  bath  robes  for 
$  480.  By  selling  \  of  them  at  $  6  each,  i  of  them  at  $  7  each, 
^  of  them  at  f  5  each,  and  the  rest,  or  i  of  them,  at  $  8  each, 
he  gained  $  128.     How  many  did  he  sell  at  each  price  ? 

14.  In  a  certain  balloon  race,  the  sum  of  the  distances 
covered  by  the  Lotus  II  and  the  United  States  was  1025  miles. 
The  distance  covered  by  the  former  was  50  miles  more  than  \ 
of  that  covered  by  the  latter.     How  far  did  each  travel  ? 

15.  A  newspaper  reporter  saved  \  of  his  weekly  salary,  or 
$  1  more  than  was  saved  by  an  artist  on  the  same  paper,  v/hose 
salary  was  $  5  greater  but  who  saved  only  ^  of  it.  How  much 
did  the  reporter  earn  per  week?  the  artist? 

16.  During  a  year  of  365  days  one  locality  had  6  days  less 
of  ^ clear'  weather  than  of  'cloudy'  weather,  and  4  days  more 
of  'clear'  than  of  'partly  cloudy'  weather.  Pind  the  num- 
ber of  days  of  each  kind  of  weather  during  the  year. 

17.  The  bark  from  a  cork  tree  lost  \  of  its  weight  by  being 
boiled.  The  boiled  cork  was  then  scraped,  its  weight  thus 
being  reduced  \.  How  much  did  the  cork  weigh  before  and 
after  these  two  operations,  if  the  entire  loss  was  16  pounds  ? 

18.  At  a  certain  depth  a  diver  saw  the  sun  as  a  reddish 
disk.  At  a  depth  25  feet  more  than  twice  this  depth  it  could 
still  be  faintly  seen.  If  darkness  occurred  on  descending  100 
feet  more,  or  at  a  total  depth  of  325  feet,  at  what  depth  did 
the  sun  appear  as  a  reddish  disk  ? 


SIMPLE  EQUATIONS  159 

19.  Find  a  fraction  whose  value  is  f  and  whose  denominator 
is  15  greater  than  its  numerator. 

20.  Find  a  fraction  whose  value  is  |  and  whose  numerator 
is  3  greater  than  half  of  its  denominator. 

21.  The  numerator  of  a  certain  fraction  is  8  less  than  the 
denominator.  If  each  term  of  the  fraction  is  decreased  by  5, 
the  resulting  fraction  equals  ^.     What  is  the  fraction  ? 

22.  An  acre  of  wheat  yielded  2000  pounds  more  of  straw 
than  of  grain.  The  weight  of  the  grain  was  .3  of  the  total 
weight  of  grain  and  straw.  How  many  60-pound  bushels  of 
wheat  were  produced? 

23.  The  total  diameter  of  a  large  wooden  fly  wheel  is 
30  feet.  The  number  of  inches  in  the  thickness  of  the  rim  is 
2  less  than  the  number  of  feet  from  the  center  to  the  rim. 
How  thick  is  the  rim  ? 

24.  A  shipment  of  83,000  postal  cards  in  two  sizes  weighed 
472  pounds.  The  smaller  cards  weighed  5  pounds  per  1000 
and  the  larger  ones  weighed  6  pounds  3  ounces  per  1000.  Find 
the  number  of  cards  of  each  size  in  the  shipment. 

25.  I  paid  18^  more  for  a  screen  door,  7  feet  by  3  feet,  than 
for  3  window  screens,  each  2|  feet  by  3  feet.  Find  the  price  per 
square  foot  in  each  case,  if  it  was  3^  less  for  window  screens. 

26.  A  grocer  bought  a  box  of  soap  containing  72  cakes  for 
$4.50.  Some  of  the  soap  he  sold  at  3  cakes  for  25^,  and  the 
rest  at  10^  a  cake.  This  gave  a  profit  of  $1.90.  How  many 
cakes  did  he  sell  at  each  price  ? 

27.  It  costs  3.6  ^  less  to  travel  100  miles  on  the  Swiss  rail- 
roads under  public  management  than  it  did  to  travel  90  miles 
when  they  were  under  private  management.  The  average  fare 
per  mile  under  private  management  was  1.9^  more  than  it  is 
under  public  management.    Find  the  rate  per  mile  under  each. 

28.  Two  orange  pickers  together  earned  $4.50  a  day,  and 
one  of  them  picked  20  boxes  more  than  the  other.  If  the 
slower  one  had  picked  twice  as  many  as  he  did,  they  would 
have  earned  $  6.50.     How  much  did  each  receive  a  box  ? 


160  SIMPLE  EQUATIONS 

29.  A  can  do  a  piece  of  work  in  8  days.  If  B  can  do  it  in 
10  days,  in  how  many  days  can  both  working  together  do  it  ? 

Solution 
Let  X  =  the  required  number  of  days. 

Then,         -  =  the  part  of  the  work  both  can  do  in  1  day, 

\  =  the  part  of  the  work  A  can  do  in  1  day, 
yiy  =  the  part  of  the  work  B  can  do  in  1  day  ; 

...1  =  1  +  1. 

X      8      10 

Solving,      X  =  4f ,  the  required  number  of  days. 

30.  A  can  do  a  piece  of  work  in  10  days,  B  in  12  days,  and 
C  in  8  days.     In  how  many  days  can  all  together  do  it  ? 

31.  It  takes  a  man  6  days  to  make  a  Panama  hat,  and  a  boy 
7  days.  How  long  would  it  take  them,  if  they  could  work  to- 
gether ? 

32.  The  average  amount  of  coal  blasted  out  by  a  keg  of 
powder  can  be  mined  by  one  man  in  2  days  and  by  another 
in  3  days.  How  long  would  it  take  them  to  mine  it  if  they 
worked  together  ? 

33.  A  and  B  can  dig  a  ditch  in  10  days,  B  and  C  can  dig  it 
in  6  days,  and  A  and  C  in  7^  days.  In  what  time  can  each 
man  do  the  work  ? 

Suggestion.  —  Since  A  and  B  can  dig  -^^  of  the  ditch  in  1  day,  B  and  C 
I  of  it  in  1  day,  and  A  and  C  j^  of  it  in  1  day,  iiy  +  i  +  i^j  is  twice  the 
part  they  can  all  dig  in  1  day. 

34.  A  and  B  can  load  a  car  in  1|  hours,  B  and  C  in  2^  hours, 
and  A  and  C  in  2^  hours.  How  long  will  it  take  each  alone 
to  load  it  ? 

35.  In  a  certain  year.  New  York  State  furnished  158.9  mil- 
lion pens,  or  54  %  of  all  that  were  made  in  the  United  States. 
How  many  pens  were  made  in  the  United  States  ? 

36.  The  per  cent  of  copper  contained  in  an  ancient  die  found 
in  Egypt  was  2^  %  more  than  3  times  the  per  cent  of  tin.  If 
these  metals  formed  92^  %  of  the  die,  what  per  cent  of  each 
did  it  contain  ? 


SIMPLE   EQUATIONS  161 

37.  Of  the  population  of  Mexico  at  one  time  the  per  cent 
of  whites  was  I  that  of  Indians.  Mixed  races  formed  5% 
more  than  the  per  cent  of  Indians.     Find  the  per  cent  of  each. 

38.  Crude  oil  when  refined  produces  2  J  times  as  much  kero- 
sene as  it  does  gasoline,  and  the  remainder,  which  is  65%,  is 
fuel  oil.  If  a  certain  refinery  produces  2250  barrels  of  kero- 
sene a  day,  what  is  its  daily  capacity  of  crude  oil  ? 

39.  The  units'  digit  of  a  two-digit  number  exceeds  the  tens' 
digit  by  5.  If  the  number  increased  by  63  is  divided,  by  the 
sum  of  its  digits,  the  quotient  is  10.     Find  the  number. 

-  Solution 

Let  X  =  the  digit  in  tens'  place. 

Then,  a;  -f-  5  =  the  digit  in  units'  place, 

and  10  a;  +  (z+  6)  =  the  number  ; 

.  10  a;  +  (x-f5)  +  63_.^Q. 
2x  +  5  ' 

whence,  x  =  2, 

and  X  -f-  6  =  7. 

Therefore,  the  number  is  27. 

40.  The  tens'  digit  of  a  two-digit  number  is  3  times  the 
units'  digit.  If  the  number  diminished  by  33  is  divided  by  the 
difference  of  the  digits,  the  quotient  is  10.     Find  the  number. 

41.  The  tens'  digit  of  a  two-digit  number  is  J  of  the  units' 
digit.  If  the  number  increased  by  27  is  divided  by  the  sum 
of  its  digits,  the  quotient  is  6J.     Find  the  number. 

42.  An  officer,  attempting  to  arrange  his  men  in  a  solid 
square,  found  that  with  a  certain  number  of  men  on  a  side  he 
had  34  men  over,  but  with  one  man  more  on  a  side  he  needed 
35  men  to  complete  the  square.     How  many  men  had  he  ? 

Suggestion.  —  With  x  men  on  a  side,  the  square  contained  x^  men  ; 
with  (x  +  1)  men  on  a  side,  there  were  places  for  (x  -f  1)^  men. 

43.  A  regiment  drawn  up  in  the  form  of  a  solid  square  was 
reenforced  by  240  men.  When  the  regiment  was  formed  again 
in  a  solid  square,  there  were  four  more  men  on  a  side.  How 
many  men  were  there  in  the  regiment  at  first  ? 

milne's  1st  yr.  alg.  —  11 


162  SIMPLE   EQUATIONS 

44.  At  what  time  between  6  and  6  o'clock  will  the  hands  of 
a  clock  be  together  ? 

Solution 

Starting  with  the  hands  in  the  position  shown, 
at  5  o'clock,  let  x  represent  the  number  of  minute 
spaces  passed  over  by  the  minute  hand  after  5 
o'clock  until  the  hands  come  together.  In  the  same 
time  the  hour  hand  will  pass  over  ^^  of  x  minute 
spaces. 

Since  they  are  25  minute  spaces  apart  at  5  o'clock, 

a;  _  £-  =  25  ; 
12 

,'.x  —  27^^,  the  number  of  minutes  after  5  o'clock. 

45.  At  what  time  between  1  and  2  o'clock  will  the  hands  of 
a  clock  be  together  ? 

46.  At  what  time  between  10  and  11  o'clock  will  the  hands 
of  a  clock  point  in  opposite  directions  ? 

47.  At  what  two  different  times  between  4  and  o  o'clock 
will  the  hands  of  a  clock  be  15  minute  spaces  apart  ? 

48.  Mr.  Reynolds  invested  $800,  a  part  at  6  %,  the  rest  at 
5%.  The  total  annual  interest  was  $45.  Find  how  much 
money  he  invested  at  each  rate. 

Suggestion.  — Let  a;  =  the  number  of  dollars  invested  at  6%. 
Then,  800  —  a;  =  the  number  of  dollars  invested  at  5%  ; 

49.  A  man  put  out  $  4330  in  two  investments.  On  one  of 
them  he  gained  12%,  and  on  the  other  he  lost  5%.  If  his 
net  gain  was  $251,  what  was  the  amount  of  each  investment? 

50.  Mr.  Bailey  loaned  some  money  at  4  %  interest,  but  re- 
ceived $  48  less  interest  on  it  annually  than  Mr.  Day,  who  had 
loaned  -f  as  much  at  6  % .     How  much  did  each  man  loan  ? 

51.  A  man  paid  $80  for  insuring  two  houses  for  $6000  and 
$  4000,  respectively.  The  rate  for  the  second  house  was  \  % 
greater  than  that  for  the  first.     What  were  the  two  rates  ? 


simplp:  equations  163 

52.  United  States  silver  coins  are  ^-^  pure  silver,  or  *  ^^  fine.' 
How  much  pure  silver  must  be  melted  with  250  ounces  of 
silver  ^  fine  to  render  it  of  the  standard  fineness  for  coinage? 

Suggestion.  —  Let  x  =  the  number  of  ounces  of  pure  silver  to  be  added. 

Then,  ^250)  -\-x=  the  number  of  ounces  of  pure  silver  after  the  addition. 

Also,  1^^(250  +  x)  =  the  number  of  ounces  of  pure  silver  after  the  addition. 

53.  In  an  alloy  of  90  ounces  of  silver  and  copper  there  are 
6  ounces  of  silver.  How  much  copper  must  be  added  that  10 
ounces  of  the  new  alloy  may  contain  |  of  an  ounce  of  silver? 

54.  If  80  pounds  of  sea  water  contain  4  pounds  of  salt,  how 
much  fresh  water  must  be  added  that  49  pounds  of  the  new 
solution  may  contain  If  pounds  of  salt  ? 

55.  Four  gallons  of  alcohol  90  %  pure  is  to  be  made  50  % 
pure.     What  quantity  of  water  must  be  added? 

56.  Of  24  pounds  of  salt  water,  12  %  is  salt.  In  order  to 
have  a  solution  that  shall  contain  4  %  salt,  how  many  pounds 
of  pure  water  should  be  added? 

57.  A  man  rows  downstream  at  the  rate  of  6  miles  an  hour 
and  returns  at  the  rate  of  3  miles  an  hour.  How  far  down- 
stream can  he  go  and  return  within  9  hours  ? 

58.  An  airship  traveled  11  miles  with  the  wind  in  the  same 
time  as  1  mile  against  it.  If  it  traveled  55  miles  and  returned 
in  12  hours,  what  was  its  rate  against  the  wind?  with  the 
wind  ? 

59.  A  train  went  905.4  miles  in  a  certain  length  of  time. 
Another  train  with  a  speed  3  miles  greater  per  hour  covered 
54  miles  more  in  the  same  length  of  time.  What  was  the 
speed  of  each  train  ? 

60.  An  express  train  whose  rate  is  40  miles  an  hour  starts  1 
hour  and  4  minutes  after  a  freight  train  and  overtakes  it  in 
1  hour  and  36  minutes.  How  many  miles  does  the  freight 
train  run  per  hour? 


164  SIMPLE   EQUATIONS 

Solution  of  Formulae 

216.  A  formula  expresses  a  principle  or  a  rule  in  symbols. 
The  solution  of  problems  in  commercial  life,  and  in  mensura- 
tion, mechanics,  heat,  light,  sound,  electricity,  etc.,  often  de- 
pends upon  the  ability  to  solve  formulae. 

EXERCISES 

217.  1.  The  circumference  of  a  circle  is  equal  to  tt  (=  3.1416) 
times  the  diameter,  or 

C  =  7rD. 

Solve  the  formula  for  D  and  find,  to  the  nearest  inch,  the 
diameter  of  the  wheel  of  a  locomotive,  if  the  circumference  of 
the  wheel  is  194.78  inches. 

Solution 

From  C  =  ttD,  ttD  =  C. 

.•.i)=^  =  lM:I§  =  62.0H-. 
TT      3.1416 

Hence,  to  the  nearest  inch,  the  diameter  is  62  inches. 

2.  Area  of  a  triangle  =  \  (base  x  altitude),  or 

A  =  \bh. 

Solve  for  6,  then  find  the  base  of  a  triangle  whose  area  is 
600  square  feet  and  altitude  40  feet. 

3.  The  area  of  a  trapezoid  is  equal  to  the  product  of  the 
altitude  and  half  the  sum  of  the  bases ;  that  is, 

A  =  h-\{b  +  bi). 

The  bases  are  h  and  6'.    6'  is  read  '  &-prime.* 

Solve  for  li,  then  find  the  altitude  of  a  trapezoid  whose  area 
is  96  square  inches  and  whose  bases  are  14  inches  and  10 
inches,  respectively. 

4.  The  volume  of  a  pyramid  =  \  (base  x  altitude),  or 

Solve  for  J5,  then  find  the  area  of  the  base  of  a  pyramid 
whose  volume  is  252  cubic  feet  and  altitude  9  feet. 


SIMPLE   EQUATIONS  165 

5.  The  charge  (c)  for  a  telegram  from  New  York  to  Chicago, 
40^  for  10  words  and  3^  for  each  additional  word,  may  be 
found  by  the  formula, 

(?  =  40  +  3(/i-10), 

in  which  n  stands  for  the  number  of  words. 
Find  the  cost  of  a  16-word  message. 
Solve  for  n,  then  find  how  many  words  can  be  sent  for  $  1. 

6.  In  the  formula,      i  =  p  -  — -^  •  f , 

i  denotes  the  interest  on  a  principal  of  p  dollars  at  simple 
interest  at  r%  for  t  years. 

Solve  for  t,  then  find  the  time  $300  must  be  on  interest  at 
5  %  to  yield  $  60  interest. 

Solve  for  r.  At  what  rate  of  interest  will  $4500  yield  $900 
interest  in  5  years  ? 

Solve  for  p.  What  principal  at  3^%  will  yield  $210 
annually? 

7.  The  formula  for  the  space  (s)  passed  over  by  a  body 
that  moves  with  uniform  velocity  {v)  during  a  given  time  {t)  is 

5  =  vi. 

Solve  for  v,  then  find  the  velocity  of  sound  when  the  condi- 
tions are  such  that  it  travels  8640  feet  in  8  seconds. 

8.  The  formula  for  converting  a  temperature  of  F  degrees 
Fahrenheit  into  its  equivalent  temperature  of  C  degrees  Centi- 
grade is 

<?  =  5(f-32). 

Solve  for  F  and  express  25°  Centigrade  (the  mean  annual 
temperature  in  Havana)  in  degrees  Fahrenheit. 
Solve : 

9.  s  =  ^  at\  for  a.  13.   Mv^  =  mVi,  for  m. 

10.  F=Ma,iova.  14.    E  =  hMv\iov  M. 

11.  TF=  Fs,  for  s.  15.    s  =  VQt  +  \  af;  for  a. 

12.  P=:PR,foTE,  16.   s  =  |a(2^-l),  for^. 


166 


SIMPLE   EQUATIONS 


17.  Any  sort  of  a  bar  resting  on  a  fixed  point  or  edge  is 
called  a  lever ;  the  point  or  edge  is  called  the  fulcrum. 

A  lever  will  just  balance  when 

the  numerical  product  of  the  power      @ F        ^ 

{p)  and  its  distance  (d)  from  the       t  ^  A  ^     f 

fulcrum  (F)  is  equal  to  the  numeri- 
cal product  of  the  weight  {W)  and  its  distance  {D)  from  the 
fulcrum ;  that  is,  when 

pd=WD. 

Solve  for  TT  and  find  what  weight  a  power  of  150  (pounds) 
will  support  by  means  of  the  lever  shown,  if  d  =  7  (feet)  and 
Z)  =  3  (feet). 

Find  for  what  values  of  p,  d,  W,  or  D  the  following  levers 
will  balance,  each  lever  being  8  feet  long : 


18. 


19. 


P                            F      lU 

20. 
21. 

600            F 

W 

\               6            A2 1            . 
300       F                           100 

■i,     3    A 

F 

id  A     8-^       i 

700 

5 

A 

22.  Philip,  who  weighs  114  pounds,  and  William,  who 
weighs  102  pounds,  are  balanced  on  the  ends  of  a  9-fcot  plank. 
Neglecting  the  weight  of  the  plank,  how  far  is  Philip  from  the 
fulcrum  ? 

23.  The  figure  illustrates  the  lever  of  a  safety  valve,  the 

power  being  the  steam  pres- 
sure (p)  acting  on  the  end  of 
the  piston  above.  The  area 
of  the  end  of  the  piston  is  16 
square  inches.  What  weight 
(  W)  must  be  hung  on  the  end 
of  the  lever  so  that  when  the 
steam   pressure  rises  to   100 

pounds  per  square  inch  the  piston  will  rise  and  allow  steam  to 
escape  ? 


■^^^.^^v^w.wk^^^^^^^w^^^^'^!!g!lsgrf 


■^v.-^il  '^^''•  iy-'-^  ■•  '^'<' '  'i :  C'  ,f -"  jJr 


SIMPLE   EQUATIONS  167 

24.  The  number  of  pounds  pressure  (P)  on  A  square  feet 
of  surface  of  any  body  submerged  to  a  depth  of  h  feet  in  a 
liquid  that  weighs  w  pounds  per  cubic  foot  is  given  by  the 

formula 

P  =  wAh. 

Fresh  water  weighs  about  62^  pounds  per  cubic  foot,  and  ordinary  sea 
water  about  64  pounds  per  cubic  foot. 

Find  the  pressure  on  1  square  foot  of  surface  at  the  bottom 
of  a  standpipe  in  which  the  water  is  30  feet  high;  at  the 
bottom  of  the  ocean  at  a  depth  of  3000  feet. 

25.  Solve  P=wAh  for  h  and  find  the  value  of  h  when 
P=  5000,  ic  =  62i,  and  A  =  S. 

26.  At  what  depth  in  fresh  water  will  the  pressure  on  an 
object  having  a  total  area  of  4  square  feet  be  2000  pounds  ? 

27.  How  deep  in  the  ocean  can  a  diver  go,  without  danger, 
in  a  suit  of  armor  that  can  sustain  safely  a  pressure  of  140 
pounds  per  square  inch  (20,160  pounds  per  square  foot)  ? 

28.  If  the  pressure  per  square  foot  on  the  bottom  of  a  tank 
holding  18  feet  of  petroleum  is  990  pounds,  what  is  the  weight 
of  the  petroleum  per  cubic  foot  ? 

29.  The  side  of  a  chest  lying  in  25  feet  of  water  was  5 
square  feet  in  area  and  sustained  a  pressure  of  8000  pounds. 
Was  the  chest  submerged  in  fresh  water  or  in  salt  water  ? 

Solve : 

30.  |  =  |,fori?.  32.    a  =  ^o,for^,. 

'''    ^=^.>^-''-  '''    ^=^{^+243)'°^^- 

34.  Solve  ^  =  ^stl\  for  E;  for  r, 

e  r 

35.  Solve  i  =  -  +  ifor  f^:  for  f,. 

36.  Solve  ^Wl  =  ^,  for  W;  for  S;  for:?^. 

c  c 


SIMULTANEOUS   SIMPLE   EQUATIONS 


TWO   UNKNOWN   NUMBERS 

218.  In  the  equation  x  +  y  =  12, 

X  and  y  may  have  an  unlimited  number  of  pairs  of  values, 

as  x  =  l  and  2/  =  11 ; 

or  x  =  2  and  ?/  =  10 ;  etc. 

For  since  y  =  12  —  x, 

if  any  value  is  assigned  to  x,  a  corresponding  value  of  y  may 
be  obtained. 

An  equation  that  is  satisfied  by  an  unlimited  number  of  sets 
of  values  of  its  unknown  numbers  is  called  an  indeterminate 
equation. 

219.  Principle.  — Any  single  equation  involving  two  or  more 
unknown  numbers  is  indeterminate. 


=  10| 

=  15  J 


220.  The  equations     2x-{-2y  =  10 

and  3  a;  4-  3  2/ 

express  but  one  relation  between  x  and  2/;  namely,  that  their 

sum  is  5.     In  fact,  the  equations  are  equivalent  to 

x  +  y  =  5 
and  to  each  other.     Such  equations  are  often  called  dependent 
equations,  for  either  may  be  derived  from  the  other. 

221.  The  equations         x  +  y  =  5] 

x  —  y  =  l} 

express  two  distinct  relations  between  x  and  y,  namely,  that 

168 


SIMULTANEOUS   SIMPLE   EQUATIONS  169 

their  sum  is  5  and  their  difference  is  1.  The  equations  cannot 
be  reduced  to  the  same  equation ;  that  is,  they  are  not  equivalent. 
Equations  that  express  different  relations  between  the  un- 
known numbers  involved,  and  so  cannot  be  reduced  to  the 
same  equation,  are  called  independent  equations. 

222.  Each  of  the  equations 

x  —  y  =  li 

is  satisfied  separately  by  an  unlimited  number  of  sets  of  values 
of  X  and  y,  but  these  letters  have  only  one  set  of  values  in  both 
equations,  namely, 

a;  =  3  and  y  =  2. 

Two  or  more  equations  that  are  satisfied  by  the  same  set  or 
sets  of  values  of  the  unknown  numbers  form  a  system  of 
simultaneous,  or  consistent,  equations. 

223.  The  equations 

y 

have  no  set  of  values  of  x  and  y  in  common. 
Such  equations  are  called  inconsistent  equations. 

224.  The  student  is  familiar  with  the  methods  of  solving 
simple  equations  involving  one  unknown  number.  The  general 
method  of  solving  a  system  of  two  independent  simultaneous 
simple  equations  in  two  unknown  numbers,  as 

x  +  y  =  5) 
x-y=^S) 

is  to  combine  the  equations,  using  axioms  1-5  (§§  68,  74)  in 
such  a  way  as  to  obtain  an  equation  involving,  x  alone,  and 
another  involving  y  alone,  which  may  be  solved  separately  by 
previous  methods. 

The  process  of  deriving  from  a  system  of  simultaneous  equa- 
tions another  system  involving  fewer  unknowa  numbers  is 
called  elimination. 


x-\-y  =  5] 


170 


SIMULTANEOUS   SIMPLE   EQUATIONS 


Elimination  by  Addition  or  Subtraction 

225.  Elimination  by  addition  or  subtraction  has  been  dis- 
cussed and  applied  to  the  solution  of  simultaneous  equations 
in  §§99-103. 

EXERCISES 

226.  1.  Solve  the  equations  2  a;  +  3  ?^  =  7  and  3  a;  +  4  ?/  =  10. 

Solution 

|2a;+    3y=7,  .                          (1) 

\sx-\-    4y  =  10.  (2) 

8a;+ 12^  =  28.  (3) 

9x  +  12?/  =  30.  (4) 

X=:2.  (5) 

4+   Sy  =  1. 
.'.y  =  l. 
To  verify,  substitute  2  for  x  and  1  for  y  in  the  given  equations. . 


(1)  X  4, 

(2)  X  3, 
(4)  -  (3), 
Substituting  (5)  in  (1), 


Rule.  —  If  necessary,  multiply  or  divide  the  equations  by  such 
numbers  as  will  make  the  coefficients  of  the  quantity  to  be  elimi- 
nated numerically  equal. 

Eliminate  by  addition  if  the  resulting  coefficients  have  unlike 
signs,  or  by  subtraction  if  they  have  like  signs. 

Solve  by  addition  or  subtraction,  and  verify  results : 

3  d  4-  4  ?/  =  25, 
4d  +  32/  =  31. 

5^  +  6g  =  32, 
7p-3q  =  22. 

f  3  a  +  6  2  =  39, 
[9a-4.z  =  51: 

6x-5y  =  SS, 

4  a;  -f-  4  2/  =  44. 


4. 


5. 


'7ic  — 5?/  = 

=  52, 

.2x-\-5y  = 

=  47. 

'Sx-h2y  = 

=  23, 

.x  +  y  =  S. 

'Sx-4.y  = 

=  7, 

.x-\-10y  = 

25. 

(2x-10y 

=  15, 

2x-4.y  = 

:18. 

7. 


SIMULTANEOUS   SIMPLE   EQUATIONS  171 

j^      |2a+3&=17,  jj      |3m  +  llri  =  67, 

I3a  +  26  =  18.  '     l5m-3n  =  5. 

Elimination  by  Comparison 

227.   If  x=S-?j,  (1) 

and  also  x  =  2  -\- y,  (2) 

by  axiom  5,  the  two  expressions  for  x  must  be  equal. 
.'.S-y  =  2  +  y. 

By  comparing  the  values  of  x  in  the  given  equations,  (1)  and 
(2),  we  have  eliminated  x  and  obtained  an  equation  involving 
y  alone. 

This  method  is  called  elimination  by  comparison. 


228.   1. 

Solve  the 

equations  2  .r  —  3 y  =  10  and  5x  -{-  2y  =  6. 

Solution 

f2a;-3y  =  10,                                                   (1) 
l5x  +  2?/=6.                                                      (2) 

From  (1), 

.  =  ^«  +  3,.                                           (3^ 

From  (2), 

^='-'''                                            (4) 

6 

Comparing  the  values  of  x  in  (3)  and  (4), 
10  +  3y^()-2y 
2  5 

Solving,  y  =  ~2. 

Substituting  —  2  for  y  in  either  (3)  or  (4),  ^ 

x  =  2. 
To  verify^  substitute  2  for  x  and  —  2  for  y  in  the  given  equations. 

Rule.  —  Find  an  expression  for  the  value  of  the  same  unknown 
member  in  each  eq^iation,  equate  the  two  expressions^  and  solve 
the  equation  thus  formed. 


172 


SIMULTANEOUS   SIMPLE   EQUATIONS 


4. 


5. 


[7x-Sy  =  lS. 


9. 


10. 


11. 


12. 


Solve  by  comparison,  and  verify  results : 

3x-2  7j  =  10, 
[x-\-y  =  70. 
(5x-\-y  =  22, 
\x  +  5y=14:. 
(2x-\-3y  =  24:, 
{5x-3y  =  lS. 
(3x-\-5y=Uy 
\2x-3y=3. 

3v-^2y  =  36, 
[5v-9y  =  23. 


2s-\-7t  =  S, 
3s  +  9t  =  9. 
4  ?i  +  6  V  =  19, 
3u-2v  =  ^. 

1 11  v'4- 5  w  =  87. 
( 4:X—  13y  =  o, 
[3x  +  lly=  -17. 
4:X  +  3y=10, 


13. 


Ll2a;-ll^  =  -10. 
4:X-l-3y  =  27. 


(lSx~3y  =  4:, 


Elimination  by  Substitution 

229.   Given  3x-{-2y  =  27,  (1) 

and  X  —  y  =^4.  (2) 

On  solving  (2)  for  x,  its  value  is  found  to  he  x  =  4=  -\-  y. 

If  4  +  2/  is  substituted  for  x  in  (1^,  3  x  will  become  3(4  +  y), 
and  the  resulting  equation 

3(44-2/)  +  2i/=27  (3) 

will  involve  y  only,  x  having  been  eliminated. 

Solving  (3),  2/  =  3. 

Substituting  3  for  y  in  (2),  x  =  7. 

This  method  is  called  elimination  by  substitution. 

Rule.  —  Find  an  expression  for  the  value  of  either  of  the  un- 
known numbers  in  one  of  the  equations. 

Substitute  this  value^for  that  unknown  number  in  the  other 
equation,  and  solve  the  residting  equation. 


SIMULTANEOUS   SIMPLE   EQUATIONS 


173 


EXERCISES 

230.    Solve  by  substitution,  and  verify  results: 


,4:y  —  x  =  14. 

(x-hy  =  10, 
I  6  ic  -  7  V  =  34. 


3. 


4. 


3ic-4?/  =  26, 

.x-Sy  =  22. 

'6y -10  x  =  U, 
,y  —  X  =  3. 

>4-l=3a', 
.5x  +  9  =  3y, 

6. 


7. 


8. 


9. 


10. 


17=3x-\-z, 
7  =  3z-2x. 


•iy  =  10-x, 
a;  =  5. 

7  2-3a;  =  18, 
z  —  5x  =  l. 


\y— 

(3-loy  =  -x, 
1  3  -f  15  2/  =  4  a;. 

l-x  =  3y, 
.3(l-x)  =  A0-y. 


231.  Three  standard  methods  of  elimination  have  been 
given.  Though  each  is  applicable  under  all  circumstances,  in 
special  cases  each  has  its  peculiar  advantages.  The  student 
should  endeavor  to  select  the  method  best  adapted  or  to  invent 
a  method  of  his  own. 

EXERCISES 

232.  Solve  by  any  method,  verifying  all  results : 
^      ra;  +  2;  =  13, 

.x  —  z  =  5. 


5. 


2. 


2/ =  10, 
y  =  6. 


4. 


{3x-\- 
U  +  3 

(4:X-{-5y  =  —2, 
1  5  a;  -f-  4  2/  =  2. 

(5x-y  =  2S, 
l3a;  +  5v  =  28. 


6. 


(x  +  3  =  y-3, 
\2(x-{-3)  =  6-y. 

(5x-y  =  12, 
\x-{-3y  =  12. 

(^2-x)  =  3y, 
\2(2-x)  =  2(y-2). 

(x  +  l)  +  (y-2)  =  7, 
(x  +  l)-{y-2)=5. 


174 


SIMULTANEOUS   SIMPLE   EQUATIONS 


Eliminate  before  or  after  clearing  of  fractions,  as  may  be 
more  advantageous ; 


9. 


10. 


aj  +  !  =  ll, 


4^5 
3^5 


11. 


12. 


X     2y_      o 


o  X 


"^^ -{-'!  =  12. 

2       3 
x-\-y      x  —  y 


2 


3 


8, 


3  4 


13.     p  +  i 


(3a;-2/-l)=i  +  f(2/-l), 
i(4a.  +  32/)  =  TV(7  2/  +  24). 


Equations  of  the  form  -4--  =  c,  though  not  simple  equa- 
X      y 

tions,  may  be  solved  as  simple  equations  for  some  of   their 

roots  by  first  regarding  -  and  -  as  the  unknown  numbers. 
X  y 


14.    Solve  the  equations   • 

4      3      14 
X     y      5 
4      10      50 

ix'^  y        3* 

Solution.     (2)  —  (1), 

13  _  208 
y      15* 

.     1      16 

"  y     15* 

Substituting  (3)  in  (1), 

4_48_14, 
X     15      5  * 

"    X     2* 

From  (4)  and  (3), 

-1— i 

(1) 

(2) 


(3) 


(*) 


SIMULTANEOUS   SIMPLE   EQUATIONS 


175 


Solve 

,  and  verify  results : 

5_§=-2 

2      3      . 
=  5, 

15.    . 

X     y 

18. 

a;     2/ 

25  .  1      ^ 

5      2_^ 

h  -  =  6. 

[x      y 

.a;     ^~    * 

[i-«=-i, 

'4,3     9 

-  +  -  =  o' 

16.    . 

X     y 

19. 

X     y     ^ 

13     25 

3,4     11 

x'^y     28' 

X    y    12 

3      1_      o 

r  5     4    o 

7r-  +  -  =  3» 

17.     ^ 

2  a;     ?/  ~~ 

20.     , 

3a;      y 

#  +-=23. 

^-f  =  2^ 

.2  a;     y 

y     6a;       ^ 

Literal  Simultaneous  Equations 

1.    Solve  the  equations    f«^  +  %  =  ^'*> 
I  ca;  -f  dy  =  w. 


(1) 

xd, 

(2) 

x6, 

(3) 

-(4) 

(1) 

xc, 

(2) 

xa, 

(7) 

-(6) 

Solution 

ax-^by=m 

cz-\-  dy  =  71 

ddx  +  bdy  =  dwi 

6ca;  +  bdy  =  bn 

(ad  —  bc)x  =  dm'~bn 

.       _(Zm  —  bn 
ad—  be* 

acz  4-  bey  =  cm 

acx  +  ady  =  an 

{ad  —  bc)y  =  an  —  cm 

.   «  =  gro  —  cm 
ad  —  be 


(1) 
(2) 
(3) 

(4) 

(6) 

(«) 

(7) 

(8) 


In  solving  literal  simultaneous  equations,  elimination  is  usually  per- 
formed most  easily  by  addition  or  subtraction. 

(i 
^  \ 
.  ^1 


176 


SIMULTANEOUS   SIMPLE   EQUATIONS 


Solve  for  x  and  y  in  exercises   2-15,  and   test   results  by 
assigning  suitable  values  to  the  other  letters : 


5. 


by  =  171, 
ay  =  c. 


(ax  + 
\bx- 
(ax  —  by  =  m, 
[ex  —  dy  =  r. 
(ax  =  by, 
\x-{-y  =  ab. 
(m{x-\-y)  =  a, 
[  n(x  —  2/)  =  2  a. 


7. 


9. 


(a(x-y)  =5, 
[bx  —  cy  =  n. 
iaia-x)  =b(y-b), 
[  ax  =  by. 


Q     ^x  +  y=-.b-a, 

I  bx  —  ay  H-  2  a6  =  0. 


(x-y  =  a-b, 
[  ax  -f  by  =  a^  —  b\ 


Eliminate  before  or  after  clearing  of  fractions  as  seems  best 


10. 


11. 


12. 


a      b 
.bx  —  ay  =  0, 

1 

a 
1 
b 


0, 


X     y 
1      1 


{x  +  l^a  +  b-\-l 
IZ/  \y  +  l      a  -  6  +  f 
[x-y  =  2b. 

1^1 
X—  a 

x  +  y 


14. 


y 


y 


---  =  -1, 

X      y 
b 


15. 


a 


ix 


=  -1. 


y 


x-y 

X  ,  y 
a     b 

lb      c 


Problems 

234.  Solve  the  following  problems  and  verify  each  solution: 
'  1.  The  length  of  a  lot  is  20  yards  more  than  its  width,  and 
its  perimeter  is  360  yards.     Find  its  dimensions. 

2.  The  best  Panama  hats  bought  in  Colombia  cost  $5 
each,  and  the  cheapest  cost  50  cents  each.  If  18  hats  cost 
$  45,  how  many  hats  of  each  kind  were  bought  ? 


SIMULTANEOUS   SIMPLE   EQUATIONS  177 

^  S.  A  grocer  sold  2  boxes  of  raspberries  and  3  of  cherries  to  one 
customer  for  54  ^,  and  3  boxes  of  raspberries  and  2  of  cherries 
to  another  for  56  ^.     Find  the  price  of  each  per  box. 

4.  A  druggist  wishes  to  put  500  grains  of  quinine  into 
3-grain  and  2-grain  capsules.  He  fills  220  capsules.  How  many 
capsules  of  each  size  does  he  fill  ? 

5.  On  the  Fourth  of  July,  850  glasses  of  soda  water  were 
sold  at  a  fountain,  some  at  5.^  each,  the  others  at  10  ^  each. 
The  receipts  were  $  55.     How  many  were  sold  at  each  price  ? 

6.  A  fruit  dealer  bought  36  pineapples  for  $  2.50.  He  sold 
some  at  12  ^  each  and  the  rest  at  10  ^  each,  thereby  gaining 
$  1.50.     How  many  did  he  sell  at  each  price  ? 

7.  The  receipts  from  300  tickets  for  a  musical  recital  were 
$  125.  Adults  were  charged  50^  each  and  children  25^  each. 
How  many  tickets  of  each  kind  were  sold  ? 

8.  A  dealer  packed  1800  Christmas  wreaths  in  6  barrels  and 
6  cases.  Later  he  packed  2250  wreaths  in  9  barrels  and  7  cases. 
Find  the  capacity  of  a  barrel ;  of  a  case. 

9.  East  African  hemp  is  worth  $25  more  per  ton  than 
Mexican  hemp.  If  2  tons  of  African  hemp  are  worth  $  25  less 
than  2|  tons  of  Mexican  hemp,  find  the  value  of  each  per  ton. 

10.  A  natural  bridge  in  Utah  has  a  span  of  60  feet  more  than 
its  height.  If  its  height  were  200  feet  less,  it  would  be  ^  of 
its  span.     Find  the  height  arid  the  span  of  the  bridge. 

11.  A  chimney  at  Bolton,  England,  is  100  feet  lower  than 
one  at  Glasgow,  and  \  of  the  height  of  the  latter  is  64  feet 
more  than  \  of  the  height  of  the  former.  Find  the  height  of 
each. 

12.  An  errand  boy  went  to  the  bank  to  deposit  some  bills  for 
his  employer.  Some  of  them  were  l-dollar  bills,  and  the  rest 
2-dollar  bills.  The  number  of  bills  was  38  and  their  value  was 
$  50.     Find  the  number  of  each. 

MILKENS    IST    YR.    ALG. 12 


178  SIMULTANEOUS   SIMPLE  EQUATIONS 

13.  A  grocer  bought  1416  oranges  of  two  sizes.  Of  one  kind 
it  took  360  oranges  to  fill  a  box  and  of  the  other  48.  If  there 
were  10  boxes  in  all,  find  the  number  of  boxes  of  each  kind. 

j^  14.  The  cost  of  firing  20  shots  from  a  Japanese  battleship 
was  $  4040.  The  shots  from  the  large  cannon  cost  $  400  each 
and  every  shot  from  the  small  cannon  cost  $  70.  How  many- 
shots  of  each  kind  were  fired  ? 

15.  In  Berlin,  Germany,  a  mason  received  80^  more  in  5  days 
than  a  painter  received  in  6  days,  each  working  10  hours  a  day. 
The  former  earned  the  same  in  9  days  as  the  latter  did  in  12 
days.     What  was  the  hourly  wage  of  each  man  ? 

16.  The  champion  National  League  baseball  team  one  year 
won  62  games  more  than  it  lost.  The  team  that  came  second 
played  154  games,  winning  16  less  than  the  first  and  losing  18 
more  than  the  first.  How  many  games  did  each  team  win  and 
how  many  did  each  lose  ? 

17.  During  a  rate  war  between  rival  steamship  lines  the  pas- 
sage for  2  immigrants  from  Bremen  to  New  York  cost  $  7.14 
less  than  the  normal  rate  for  1,  and  the  passage  for  i  immi- 
grants ^4.76  less  than  the  normal  cost  for  3.  Find  the  nor- 
mal rate  and  the  reduced  rate. 

18.  A  man  noticed  that  a  15-word  message  by  telegraph  cost 
him  40^  and  a  22- word  message  54^,  between  the  same  two 
cities.  Find  the  charge  for  the  first  10  words  and  the  charge 
for  each  additional  word. 

19.  The  United  States  imported  38  million  bunches  of 
bananas  one  year.  The  cost,  at  30  ^  each  for  the  larger  bunches 
and  20  ^  each  for  the  smaller  ones,  was  8.55  million  dollars 
How  many  bunches  of  each  size  were  imported  ? 

20.  A  steam  pipe  was  inclosed  in  a  wooden  case.  The  diam- 
eter of  the  pipe  was  |  of  the  diameter  of  the  case.  The  radius 
of  the  case  was  2  inches  less  than  the  diameter  of  the  pipe. 
What  was  the  diameter  of  each  ? 


SIMULTANEOUS   SIMPLE    EQUATIONS  179 

21.  A  man  invested  $4000,  a  part  at  5  %  and  the  rest  at  4  % . 
If  the  annual  income  from  both  investments  was  $  175,  what 
was  the  amount  of  each  investment  ? 

22.  At  a  factory  where  1000  men  and  women  were  em- 
ployed, the  average  daily  wage  was  $  2.50  for  a  man  and  $  1.50 
for  a  woman.  If  labor  cost  $  2340  per  day,  how  many  men 
were  employed  ?  how  many  women  ? 

23.  It  required  60  inches  of  tape  to  bind  the  four  edges  of  a 
card  on  which  a  photograph  was  mounted.  The  length  of  the 
card  was  6  inches  greater  than  the  width.  How  many  inches 
long  was  the  card  ?  how  many  inches  wide  ? 

24.  The  German  railroads  carried  153  million  first-class  and 
second-class  passengers  one  year.  The  number  would  have 
been  180  million  if  4  times  as  many  had  traveled  first  class. 
How  many  traveled  first  class?  second  class? 

25.  In  one  hour  1375  vehicles  passed  a  merchant's  door  on 
Broadway,  New  York  City.  The  horse-drawn  vehicles  would 
have  equaled  the  automobiles  in  number,  had  there  been  50 
more  of  the  former  and  25  less  of  the  latter.  How  many  of 
each  passed  his  door  ? 

[^'  26.  Probably  the  highest  dock  in  the  world  is  on  the  Victoria 
Nyanza.  Its  height  above  sea  level  is  50  feet  less  than  15 
times  its  length,  and  the  sum  of  its  height  and  length  is  3950 
feet.     Find  its  height  above  sea  level. 

27.  The  American  and  British  tourists  to  Japan  during  a  re- 
cent year  numbered  2794.  If  there  had  been  twice  as  many 
Americans  and  3  times  as  many  British,  the  number  would  have 
been  6682.    How  many  tourists  were  there  from  each  country  ? 

28.  The  number  of  students  attending  the  University  of 
Berlin  at  one  time  was  9277  more  than  the  number  attending 
at  Munich,  and  ^  the  number  at  Berlin  plus  ^  the  number  at 
Munich  equaled  9591.     How  many  attended  each  university  ? 


180  SIMULTANEOUS   SIMPLE   EQUATIONS 

29.  UDder  the  present  contract,  it  costs  $  24.15  less  a  year 
per  lamp  to  maintain  electric  lights  in  a  certain  city  than  it  did 
under  the  previous  one,  and  the  expense  of  6  lamps  then  was 
$46.35  more  than  that  of  7  lamps  now.  Find  the  present 
yearly  expense  per  lamp. 

30.  On  the  same  day  two  seats  in  the  New  York  Stock 
Exchange  sold  for  $  192,500.  If  one  of  them  had  sold  for 
$  1500  less,  and  the  other  for  $  1000  more,  the  prices  of  the 
two  would  have  been  equal.     Find  the  price  of  each. 

31.  A  man  had  10  fox  skins,  some  of  which  were  silver  fox, 
worth  $300  a  pelt,  and  some  black  fox,  worth  $750  a  pelt. 
If  he  had  had  3  less  of  the  former  and  3  more  of  the  latter, 
the  total  value  would  have  been  $6150.  How  many  had  he 
of  each  ? 

32.  The  quantity  of  peanuts  raised  in  the  United  States  in 
a  year  is  135  million  pounds  more  than  the  quantity  of  all 
nuts  imported,  and  ^  of  the  former  equals  ^  of  the  latter. 
Find  the  number  of  pounds  of  nuts  imported  and  of  peanuts 
raised  in  the  United  States. 

33.  The  receipts  from  a  football  game  were  $  700.  Admis- 
sion tickets  to  the  grounds  were  sold  for  50  ^,  and  to  the  grand 
stand  for  25  ^  in  addition.  If  twice  as  many  persons  had  pur- 
chased tickets  for  the  grand  stand,  the  receipts  would  have 
been  $  800.     How  many  tickets  of  each  kind  were  sold  ? 

34  A  train  of  25  cars  loaded  with  iron  ore  wa§  run  out  on  a 
dock  and  the  ore  emptied  into  pockets  beneath  the  tracks. 
The  ore  filled  7  pockets  and  ^  of  another.  To  fill  this  last 
pocket,  then,  required  16  tons  less  than  2  extra  car  loads. 
What  was  the  capacity  of  a  car?    of  a  pocket? 

35.  If  100  pounds  of  soft  coal  in  burning  can  evaporate  50 
pounds  more  water  than  6  gallons  of  oil,  and  if  60  pounds  of 
coal  can  evaporate  10  pounds  less  water  than  4  gallons  of  oil, 
how  many  pounds  of  water  can  1  pound  of  coal  evaporate  ? 
1  gallon  of  oil  ? 


SIMULTANEOUS   SIMPLE   EQUATIONS  181 

36.  A  proposed  tunnel  under  Bering  Strait  would  be  in 
three  sections,  each  of  which  would  be  J  of  a  mile  longer,  and 
two  of  which  together  would  be  12f  miles  longer  than  the 
Simplon  tunnel.  Find  the  length  of  the  proposed  tunnel ;  of 
the  Simplon  tunnel. 

37.  If  1  is  added  to  the  numerator  of  a  certain  fraction,  the 
value  of  the  fraction  becomes  f ;  if  2  is  added  to  the  denom- 
inator, the  value  of  the  fraction  becomes  |.  What  is  the 
fraction  ? 

Suggestion.  —  Let  x  =  the  numerator  and  y  =  the  denominator. 

38.  If  the  numerator  of  a  certain  fraction  is  decreased  by  2, 
the  value  of  the  fraction  is  decreased  by  ^ ;  but  if  the  denomi- 
nator is  increased  by  4,  the  value  of  the  fraction  is  decreased 
by  |.     What  is  the  fraction  ? 

39.  A  certain  number  expressed  by  two  digits  is  equal  to  7 
times  the  sum  of  its  digits ;  if  27  is  subtracted  from  the  num- 
ber, the  difference  will  be  expressed  by  reversing  the  order  of 
the  digits.     What  is  the  number  ? 

Suggestion.  — The  sum  of  x  tens  and  y  units  is  (lOx  -f  y)  uAits  ;  of 
y  tens  and  x  units,  (10  j/  +  x)  units. 

40.  Find  a  number  that  is  3  greater  than  6  times  the  sum 
of  its  two  digits,  if  the  units'  digit  is  2  less  than  the  tens'  digit. 

41.  A  crew  can  row  8  miles  downstream  and  back,  or  12 
miles  downstream  and  halfway  back  in  1 J  hours.  What  is  their 
rate  of  rowing  in  still  water  and  the  velocity  of  the  stream  ? 

42.  A  man  rows  12  miles  downstream  and  back  in  11  hours. 
The  current  is  such  that  he  can  row  8  miles  downstream  in 
the  same  time  as  3  miles  upstream.  What  is  his  rate  of  row- 
ing in  still  water,  and  what  is  the  velocity  of  the  stream  ? 

43.  A  quantity  of  wheat  could  be  thrashed  by  two  machines 
in  6  days,  but  the  larger  machine  worked  alone  for  8  days  and 
was  then  replaced  by  the  smaller,  which  finished  in  3  days. 
How  long  would  it  have  taken  the  larger  machine  to  thrash  all 
of  the  wheat  ?  the  smaller  machine  ? 


182  SIMULTANEOUS   SIMPLE   EQUATIONS 


THREE   UNKNOWN  NUMBERS 

235.  The  student  has  been  solving  systems  of  two  independ- 
ent simultaneous  equations  involving  two  unknown  numbers. 
In  general, 

Principle. — Every  system  of  independent  simidtaneous  simple 
equations^  involving  the  same  number  of  unknown  numbers  as 
there  are  equations,  can  be  solved,  and  is  satisfied  by  one  and 
only  one  set  of  values  of  its  unknown  numbers. 

EXERCISES 


236.    1.    Solve  . 


x-\-2y-{-3z  =  U,  (1) 

2x  +  y-\-2z  =  10,  (2) 

3x-\-Ay-Sz  =  2.  (3) 


Solution.  — Eliminating  z  by  combining  (1)  and  (3), 

(1)  +  (3),  4x+ey  =  16.  (4) 
Eliminating  z  by  combining  (2)  and  (3), 

(2)  X  3,  6  a;  4-    3  y  +  6  2  =  30 

(3)  X  2,  6x+    Sy-6z=   4 

Adding,  12x  +  ny  =34,  (5) 

Eliminating  x  by  combining  (5)  and  (4), 

(4)  X  3,  12x  +  lSy  =  48  (6) 
(6)-(5),                                          ly          =U;.:y  =  2. 

Substituting  the  value  of  ?/  in  (4),  4  cc  +  12  =  16 ;  .-.  x  =  \. 

Substituting  the  values  of  x  and  y  in  (1), 

1  +4  +  3;5  =  14;   .'.  z  =  S. 

Verification.  —  Substituting  x=l,  y  =  2,   and   2:  =3  in  the  given 
equations, 

(1)  becomes     1  +  4  4- 9  =  14,  or  14  =  14 

(2)  becomes  2  +  2  +  6  =  10,  or  10  =  10 
and  (3)  becomes  3  +  8-9=  2,  or  2=  2 
that  is,  the  given  equations  are  satisfied  for  a;  =  1,  ?/  =  2,  and  «  =  8. 


SIMULTANEOUS   SIMPLE   EQUATIONS 


183 


Solve,  and  verify  all  results 

x-\-3y  —  z  —  10, 
2.    ^  2  a;  +  5  2/  +  4  2;  =  57, 
3  a;  -  1/ +  2  z  =  15. 
x  +  y-{-z  =  53, 
x  +  2y  +  3z  =  105, 
a;  +  32/  +  4z  =  134. 
x-y  +  z  =  30, 
3y-x-z  =  12, 
7z  —  y-j-2x  =  141. 
'Sx-5y-^2z  =  5S, 
5.    lx-{-y  —  z  =  9, 

13x-9y-{-Sz  =  71. 


6. 
7. 
8. 
9. 


x-\-Sy-{-4:Z  =  SSy 
x-\-y-j-z  =  29, 
6x-\-Sy  +  Sz  =  156. 
3x-2y  +  z  =  2, 
2a:  +  5y  +  2z  =  27, 
x-{-3y  +  3z  =  25. 
x  +  iy  +  iz  =  32, 
ixA-iy-{-iz  =  15, 
ix  +  \y-\-iz  =  12. 

ix-iy  +  iz  =  5. 


10.  There  are  three  numbers  such  that  the  sum  of  ^  of  the 
first,  ^  of  the  second,  and  \  of  the  third  is  12  ;  of  J  of  the  first, 
J  of  the  second,  and  -^  of  the  third  is  9;  and  the  sum  of  the 
numbers  is  38.     What  are  the  numbers? 

11.  A,  B,  and  C  have  certain  sums  of  money.  If  A  gives  B 
$100,  they  will  have  the  same  amount;  if  A  gives  C  $100,  C 
will  have  twice  as  much  as  A;  and  if  B  gives  C  $100,  C  will 
have  4  times  as  much  as  B.     What  sum  has  each  ? 

12.  A  quantity  of  water  sufficient  to  fill  three  jars  of  differ- 
ent sizes  will  fill  the  smallest  jar  4  times ;  the  largest  jar  twice 
with  4  gallons  to  spare ;  or  the  second  jar  3  times  with  2  gal- 
lons to  spare.     What  is  the  capacity  of  each  jar? 

13.  A  contractor  used  3  scows  to  convey  sand  from  his 
dredge  to  the  dumping  ground.  He  was  credited  by  the  in- 
spector for : 

April  20,  scows  a,  6,  c,  a,  &,  c,  a,  and  6,  8  loads,  3230  cu.  yd. 
April  21,  scows  c,  a,  6,  c,  a,  &,  and  c,  7  loads,  2820  cu.  yd. 
April  22,  scows  a,  &,  c,  a,  6,  c,  and  a,      7  loads,  2870  cu.  yd. 

Eind  the  capacity  of  each  scow. 


GRAPHIC   SOLUTIONS 


SIMPLE   EQUATIONS 

237.  When  related  quantities  in  a  series  are  to  be  compared, 
as,  for  instance,  the  population  of  a  town  in  successive  jenrs, 
recourse  is  often  had  to  a  method  of  representing  quantities 
by  lines.     This  is  called  the  graphic  method. 

By  this  method,  quantity  is  photographed  in  the  process  of 
change.  The  whole  range  of  the  variation  of  a  quantity,  pre- 
sented in  this  vivid  pictorial  way,  is  easily  comprehended  at  a 
glance ;  it  stamps  itself  on  the  memory. 

238.  In  Fig.  1  is  shown  the  population  of  a  town  throughout 

its  variations  during  the  first 
13  years  of  the  town's  exist- 
ence. 

The  population  at  the  end  of 
2  years,  for  example,  is  repre- 
sented by  the  length  of  the 
heavy  black  line  drawn  upward 
from  2,  and  is  4000 ;  the  popu- 
lation at  the  end  of  6  years  is 
7000;  at  the  end  of  10  years, 
6300^  approximately;  and  so 
Fig.  1.  on. 

239.  Every  point  of  the  curved  line  shown  in  Fig.  1  exhibits 
a  pair  of  corresponding  values  of  two  related  quantities,  years 
and  population.  For  instance,  the  position  of  E  shows  that 
the  population  at  the  end  of  4  years  was  6000. 

Such  a  line  is  called  a  graph. 

184 


^ 

/ 

» 

/ 

s. 

- 

u 

/ 

3 

/ 

S 

4 

1^ 

y 

g 

^ 

? 

^ 

*  i 

^  I 

S 

i  ( 

0 

1   '. 

i    4 

>  e 

Ye 

i   s 
are 

'   1 

0  1 

1  12 

13  1 

4  1 

5 

UrilVERSITY 

OF 


GRAPHIC   SOLUTIONS 


185 


Graphs  are  useful  in  numberless  ways.  The  statistician  uses  them  to 
present  information  in  a  telling  way.  The  broker  or  merchant  uses  them 
to  compare  the  rise  and  fall  of  prices.  The  physician  uses  them  to  record 
the  progress  of  diseases.  The  engineer  uses  them  in  testing  materials  and 
in  computing.  The  scientist  uses  them  in  his  investigations  of  the  laws 
of  nature.  In  short,  graphs  may  be  used  whenever  two  related  quantities 
are  to  be  compared  throughout  a  series  of  values. 

240.  The  graph  in  Fig.  2  represents  the  rate  in  gallons  per 
day  per  person  at  which  water  was  used  in  New  York  City 
during  a  certain  day  of  24  hours. 


- 

~^ 

-^ 

120 

i 

/ 

"~ 

^ 

/ 

s, 

^ 

V 

— 

-I 

y 

\ 

K 

W) 

y 

\ 

^ 

.^ 

80 
70 
60 

->i 

■& 

§ 

1 

30 

n 

ba 

'afj-on 

iK 

id 

light 

0       1    2 
Midnight 


4    5    0 
0  A.M. 


0  12 

Koon 

Fm.  2. 


18  21  24 

0  P.M.        -^     Midnight 


Thus,  if  each  horizontal  space  represents  1  hour  (from  mid- 
night) and  each  vertical  space  10  gallons,  at  midnight  water 
was  being  used  at  the  rate  of  about  84  gallons  per  day  per  per- 
son; at  6  A.M.,  about  91  gallons;  at  1  p.m.,  the  13th  hour, 
about  108  gallons ;  etc. 

1.  What  was  the  approximate  consumption  of  water  at 
2  A.M.  ?  at  noon  ?  at  1 :  30  p.m.  ?  at  3  p.m.  ?  at  6  p.m.  ? 

2.  What  was  the  maximum  rate  during  the  day  ?  the  mini- 
mum rate  ?  at  what  time  did  each  occur  ? 

3.  During  what  hours  was  the  rate  most  uniform  ?  What 
was  the  rate  at  the  middle  of  each  hour  ? 

4.  What  was  the  average  increase  per  hour  between  6  a.m. 
and  8  a.m.  ?  the  average  decrease  between  4  p.m.  and  8  p.m.  ? 


186 


GRAPHIC   SOLUTIONS 


241.  The  graphs  in  Fig.  3  present  to  the  eye  in  a  forceful 
way  the  remarkable  contrasts  in  the  months  of  July  of  the 
years  1901  and  1904  by  showing  the  average  daily  maximum 
temperature  for  ten  cities  that  cover  the  part  of  the  United 
States  east  of  the  Eocky  Mountains. 


-/-^^ 

t       \ 

7S 

z        \± 

.    y\              1    \  ,^  N        u 

i  v           /    V 

-  VA          -\   A- 

I    I            At 

-  y         avX     j 

S    L    J           t 

7k                   Z    a    7    1 

,^              V7              3- 

sL^                            \    f 

1                 \T              7 

^ 

I                       t: 

^ 

i                             ^   - 

:      ^ 

!  o                        7   ^Z 

L 

;i                 L 

A                 4  ^ 

.  ^           -I 

^                 t     ' 

«                   7 

\s              r 

7    t       ^ 

V-.Z 

t    X    7 

\7 

Xl 

»            -\t 

'                                                                    bays  of 

the  Mcjnth 

July 
1901 


July 
1904 


12  3   4   5    6    7 


9  10  11  12  13  14 15  16  17  IS  IS  20  2L22  ^  %k^  26  27  28  29  30  31 

Fig.  3. 

The  vertical  spaces  for  0°  to  76°  inclusive  are  omitted.  In  the  follow- 
ing *  temperature '  means  'average  maximum  temperature.' 

1.  In  which  year  was  the  month  of  July  the  hotter  ? 

2.  On  how  many  days  of  July,  1901,  was  the  temperature 
helow  90°  ?  How  many  days  of  July,  1904,  had  a  temperature 
ahove  90°  ? 

3.  What  was  the  highest  temperature  for  July,  1901,  and  on 
what  day  did  it  occur?  for  July,  1904?  Give  the  lowest 
temperature  for  July  of  each  of  these  years  and  the  date  of  its 
occurrence. 

4.  Which  year  had  the  smaller  range  of  temperature  for 
July?  How  many  degrees  hotter  was  the  Fourth  of  July, 
1901,  than  the  Fourth  of  July,  1904  ? 

5.  Find  the  difference  between  the  highest  temperature  of 
July,  1901,  and  the  lowest  temperature  of  July,  1904. 


GRAPHIC   SOLUTIONS 


187 


242.  Fig.  4  gives  two  graphs,  —  one  showing  the  height  of 
water  above  zero  of  the  gage  in  the  Cumberland  Eiver  at  Nash- 
ville, the  other  showing  the  same  thing  for  the  Arkansas  Kiver 
at  Little  Rock,  from  daily  observations  taken  in  September 
of  the  year  1908. 


2    3    i    5   6    7    8   0  10  n  12  13  U  15  16  17  18  19  ^JU.  22;i3212526:i7ii82930 

Fig.  4. 

In  giving  heights  read  the  graphs  to  the  nearest  tenth  of  a  foot. 

1.  When  was  the  water  in  the  Cumberland  highest  ?  lowest? 
What  was  the  maximum  height  ?  the  minimum  ?  the  range 
between  them  ? 

2.  How  many  days  later  than  the  Cumberland  was  the 
Arkansas  at  its  maximum  ?  How  many  days  earlier  than  the 
Cumberland  was  the  Arkansas  at  its  minimum  ? 

3.  What  was  the  range  for  the  Arkansas  during  the  month  ? 

4.  State  the  difference  between  the  maximum  readings  for 
the  two  rivers ;  between  their  minimum  readings. 

5.  On  what  days  were  the  readings  higher  for  the  Arkansas 
than  for  the  Cumberland  ? 

6.  What  part  of  the  month  shows  the  greatest  and  most 
rapid  changes  in  the  height  of  the  Cumberland  ?  the  Arkansas  ? 

7.  Which  river  had  the  least  variation  in  height  during  the 
last  half  of  the  month  ? 

8.  Give  the  time  of  the  greatest  change  in  a  single  day  for 
either  river.     How  much  was  this  change  ? 


188 


GRAPHIC   SOLUTIONS 


EXERCISES 

243.  1.  Letting  each  horizontal  space  represent  10  years  and 
each  vertical  space  1  million  of  population,  locate  points  from 
the  pairs  of  corresponding  values  (years  and  millions  of  popula- 
tion given  below)  and  connect  these  points  with  a  line,  thus 
constructing  a  population  graph  of  the  United  States : 

1810,  7.2 ;  1820,  9.6 ;  1830, 12.8 ;  1840, 17.1 ;  1850,  23.2 ;  1860, 
31.4;  1870,  38.6;  1880,  50.2;  1890,  62.6;  1900,  76.3. 

2.  From  the  graph  of  exercise  1,  tell  the  period  during  which 
the  increase  in  the  population  was  greatest ;  least. 

3.  The  average  price  of  tin  in  cents  per  pound  for  the  months 
of  a  certain  year  was  :  Jan.,  23.4;  Feb.,  24.7;  Mar.,  26.2  ;  Apr., 
27.3;  May,  29.3;  June,  29.3;  July,  28.3;  Aug.,  281 ;  Sept., 
26.6 ;  Oct.,  25.8 ;  Nov.,  25.4 ;  Dec,  25.3. 

Draw  a  graph  to  show  the  variation  in  the  price  of  tin  during 
the  year  with  each  horizontal  space  representing  1  month  and 
each  vertical  space  1  cent. 

4.  Construct  the  graph  of  exercise  3,  letting  each  horizontal 
space  represent  1  cent  and  each  vertical  space  1  month. 

244.  Let  X  and  y  be  two  algebraic  quantities  so  related  that 
?/  =  2.T  — 3.  It  is  evident  that  we  may  give  x  a  series  of 
values,  and  obtain  a  corresponding  series  of  values  of  y ;  and 

that  the  number  of  such  pairs  of 
values  of  x  and  y  is  unlimited.  All 
of  these  values  are  represented  in 
the  graph  of  y  =  2x  —  Z.  Just  as 
in  the  preceding  illustrations,  so  in 
the  graph  of  y  =  2x  —  ^,  Fig.  5, 
values  of  x  are  represented  by  lines 
laid  off  on  or  parallel  to  an  x-axis, 
X'X,  and  values  of  y  by  lines  laid 
off  on  or  parallel  to  a  y-axis,  Y'Y, 
usually  drawn  perpendicular  to  the 
Fig.  5.  aj-axis. 


GRAPHIC   SOLUTIONS 


189 


For  example,  the  position  of  P  shows  that  t/  =  3  when  a;  =  3 ; 
the  position  of  Q  shows  that  y  =  o  when  a;  =  4 ;  the  position  of 
R  shows  that  ?/  =  7  when  x  =  6\  etc. 

Evidently  every  point  of  the  graph  gives  a  pair  of  corre- 
sponding values  of  x  and  y. 

245.  Conversely,  to  locate  any  point  with  reference  to  two 
axes  for  the  purpose  of  representing  a  pair  of  corresponding 
values  of  x  and  ?/,  the  value  of  x  may  be  laid  off  on  the  a;-axis 
as  an  x-distance,  or  abscissa,  and  that  of  y  on  the  y-axis  as  a 
y-distance,  or  ordinate.  If  from  each  of  the  points  on  the  axes 
obtained  by  these  measurements,  a  line  parallel  to  the  other 
axis  is  drawn,  the  intersection  of  these  two  lines  locates  the 
point. 

Thus,  in  Fig.  6,  to  represent  the  corresponding  values  a;  =  3,  y  =  3,  a 
point  P  may  be  located  by  measuring  3  units  from  0  to  ilf  on  the  jc-axis 
and  3  units  from  O  to  iVon  the  y-axis,  and  then  drawing  a  line  from  M 
parallel  to  O  T,  and  one  from  JV  parallel  to  OX,  producing  these  lines 
until  they  intersect. 

246.  The  abscissa  and  ordinate  of  a  point  referred  to  two 
perpendicular  axes  are  called  the  rectangular  coordinates,  or 
simply  the  coordinates,  of  the  point. 

Thus,  in  Fig.  5,  the  coordinates  of  P  are  OM{  =  NF)  and  MP{=  ON). 

247.  By  universal  custom  positive  values  of  x  are  laid  off 
from  0  as  a  zero-point,  or  origin,  toward  the  right,  and  neg- 
ative values  toward  the  lejl.  Also 
positive  values  of  y  are  laid  off  up- 
ivard  and  negative  values  dowmvard. 

The  point  A  in  Fig.  0  may  be 
designated  as  '  the  point  (2,  3),'  or 
by  the  equation  A  =  (2,  3). 

Similarly, 
B=(-2,  4),   C=(-;!,  -1),  and 
i)=(l,  -2). 

77ie  abscissa  is  always  written  first. 


Y 

^^H 

^ ,^ 

2— 

-3-2-10     12     3    4 

-^^-tr 

Illlllr^IIIII 


Fig.  0. 


190 


GRAPHIC    SOLUTIONS 


248.   Plotting  points  and  constructing  graphs. 

EXERCISES 

Note.  —  The  use  of  paper  ruled  in  small  squares,  called  coordinate 
paper,  is  advised  in  plotting  graphs. 

Draw  two  axes  at  right  angles  to  each  other  and  locate : 


1.  A  =  {3,  2). 

2.  B  =  (3,  -  2). 

3.  0=(4,  3). 

4.  i)  =  (4,  -3). 


9.    i  =  (0,  4). 

10.  M=(0,  -5). 

11.  xY=(3,  0). 

12.  F  =(-6,  0). 
the 


5.  ^  =  (5,5). 

6.  F  =  (-5,5). 

7.  G=(-2,5). 

8.  //=(-3,  -4). 

13.  Where  do  all  points   having   the   abscissa  0  lie? 
ordinate  0  ? 

14.  What  are  the  coordinates  of  the  origin? 

16.    Construct  the  graph  of  the  equation  2  y  —  x  =  2. 

Solution 

Solving  for  y,  y=l{x-\-2). 

Values  are  now  given  to  x  and  corresponding  values  are  computed 
for  y  by  means  of  this  equation.  The  numbers  substituted  for  x  need  not 
be  large.  Convenient  numbers  to  be  substituted  for  x  in  this  instance 
are  the  even  integers  from  —  6  to  +  6. 

When  x  =  —  6,  y=  —  2.     These  values  locate  the  point  A  =  (  —  6,  —2). 

"When  X  =  —  4,  y  =  —  1.     These  values  serve  to  locate  5  =  (—  4,  —1). 

Other  points  may  be  located  in  the  same  way. 

A  record  of  the  work  should  be  kept  as  follows : 

y=i(x-\-2) 


Y 

- 

A 

X 

^\ 

y^ 

G 

■'1 

^ 

^ 

<- 

^ 

E 

V 

X' 

^ 

^ 

D 

X 

<^ 

> 

c 

^ 

^ 

B 

y* 

A 

Y' 

. 

Fig.  7. 


X 

y 

Point 

-6 

-2 

A 

-4 

-  1 

B 

-2 

0 

C 

0 

1 

D 

2 

2 

E 

4 

3 

F 

6 

4 

a 

A  line  drawn  through  A,  B,  C,  D,  etc.,  is  the  graph  of  2  ?/  -  ^  —  2, 


GRAPHIC   SOLUTIONS  191 

Construct  the  graph  of  each  of  the  following : 
16.   y  =  3x  —  7.  19.   3a;— 2/ =  4.  22.   3a;  =  2 y. 

n.   y  =  2x  +  l.  20.   4a;  — 2/ =  10.  23.   2x-{-y  =  l. 

18.   y=2x-l.  21.    x-2y  =  2.  24.   2x-\-Sy  =  6. 

249.  It  can  be  proved  by  the  principle  of  the  similarity  of 
triangles  that: 

Principle.  —  The  graph  of  a  simple  equation  is  a  straight  line. 
For  this  reason  simple  equations  are  sometimes  called  linear 
equations. 

250.  Since  a  straight  line  is  determined  by  two  points,  to 
plot  the  graph  of  a  linear  equation,  ^j^o^  two  points  and  draw 
a  straight  line  through  them. 

It  is  often  convenient  to  plot  the  points  where  the  graph 
intersects  the  axes.  To  find  where  it  intersects  the  a;-axis,  let 
2/  =  0  ;  to  find  where  it  intersects  the  2/-axis,  let  x  =  0. 

Thus,  in  y=^(x  +  2),  when  y  =  0,x  =  -2,  locating  C7,  Fig.  7;  when 
X  =  0,  y  =  1 ,  locating  D. 

Draw  a  straight  line  through  C  and  D. 

If  the  equation  lias  no  absolute  term,  x  =  0  when  y  =  0,  and  this 
method  gives  only  one  point.  In  any  case  it  is  desirable,  for  the  sake  of 
accuracy^  to  plot  points  some  distance  apart,  as  A  and  G,  in  Fig.  7. 

EXERCISES 

251.  Construct  the  graph  of  each  of  the  following: 


1. 

y  =  x-2.     . 

8. 

2x-Sy  =  6. 

15. 

8  a;  -  3  2/  =  -  6. 

2. 

y  =  2-x. 

9. 

2x-\-3y  =  0. 

16. 

-2a; +  2/ =-3. 

3. 

y  =  9  —  4:X. 

10. 

a; -42/ =  3. 

17. 

—  3  a; +  4^  =  8. 

4. 

2/  =  4a; -9. 

11. 

7  X  —  y  =  14:. 

18. 

5x-hSy  =  7i. 

5. 

2/ =  10 -2a;. 

12. 

4-a;  =  2y. 

19. 

x-iy  =  3. 

6.   y  =  2x-10.      13.   3a; +  42/ =  12.       20.   ix-{-ly  =  2. 
1.  y  =  2x  —  4.        14.   5a;— 2 2/ =  10.       21.    .7 a;  —  .3 2/ =  .4. 


192 


GRAPHIC   SOLUTIONS 


~~ 

n 

r 

\ 

/ 

c 

\ 

nX] 

Y 

D 

\ 

< 

'/ 

u 

\ 

V 

N 

/ 

\ 

H 

/ 

\ 

V 

■R 

/ 

\ 

* 

/ 

\ 

r~ 

/ 

A 

0 

SM 

\ 

/ 

\ 

/ 

252.   Graphic  solution  of  simultaneous  linear  equations. 

1.  Let  it  be  required  to  solve  graphically  the  equations 

y^2^x,  (1) 

2/  =  6-^.  (2) 

As  in  §  248,  construct  the 
graph  of  each  equation,  as  shown 
in  Fig.  8. 

1.  When  aj  =  —  1,  the  value 
of  y  in  (1)  is  represented  by 
AB,  and  in  (2)  by  AC. 

Therefore,  when  it;  =  —  1,  the 
equations  are  not  satisfied  by 
^^'    ■  the  same  values  of  y. 

2.  Compare  the  values  of  y  when  ic  =  0 ;  when  a;  =  1 ;  2. 

3.  For  what  value  of  x  are  the  values  of  y  in  the  two  equa- 
tions equal,  or  coincident  ? 

4.  What  values  of  x  and  y  will  satisfy  hoth  equations  ? 
The  required  values  of  x  and  y^  then,  are  represented  graphi- 
cally by  the  coordinates  of  P,  the  intersection  of  the  grajyhs. 

II.   Let  the  given  equations 
x  +  y==7, 
2x  +  2y  =  lL 

5.  What  happens  if  we  try 
to  eliminate  either  a;  or  ?/  ? 

6.  Since  y  =  7  —  x  in  both 
equations,  what  will  be  the 
relative  positions  of  any  two 
points  plotted  for  the  same 
value  of  a;?  the  relative  posi- 
tions of  the  two  graphs  ? 

7.  The   algebraic  analysis   shows   that   the    equations    are 
indeterminate. 

The  graphic  analysis  also  shows  that  the  equations  are  inde- 
terminate, for  their  graphs  coincide. 


be 


\ 

\ 

\ 

4\ 

\1^ 

">« 

N 

^ 

V, 

><r.^ 

°^ 

'v 

.^^ 

^^ 

* 

^- 

\ 

\ 

\ 

- 

_^ 

Fig.  9. 


GRAPHIC   SOLUTIONS 


193 


" 

• 

Y 

- 

\ 

\ 

\ 

s 

\ 

\ 

\ 

\ 

\ 

^-y 

^ 

t.l\ 

f. 

\ 

V 

\ 

X 

\ 

\ 

X 

\ 

\ 

N 

\ 

\ 

Y 

III.    Let  the  given  equations 

be  1^  =  ^-^'  W 

[y  =  4:-x.  (2) 

8.  When  x=  —  1,  how  much 
greater  is  the  value  of  y  in  (1) 
than  in  (2),  as  shown  both  by 
the  equations  and  their  graphs  ? 

9.  Compare  the  y's  for  other 
values  of  x.  yig.  lo. 

10.  For  every  value  of  x  the  values  of  y  in  the  two  equa- 
tions differ  by  2,  and  the  graphs  are  2  units  apart,  vertically. 

In  algebraic  language,  the  equations  cannot  be  simultaneous ; 
that  is,  they  are  inconsistent. 

In  graphical  language,  their  graphs  cannot  intersect,  being 
parallel  straight  lines. 

253.  Principles.  —  1.  A  single  linear  equation  involving  two 
unknown  numbers  is  indeterminate. 

2.  Two  linear  equations  involving  tivo  tinknown  numbers 
are  determinate,  provided  the  equations  are  independent  and 
simultaneous. 

TJiey  are  satisfied  by  one,  and  only  one,  pair  of  common  values. 

3.  The  pair  of  common  values  is  represented  graphically  by 
the  coordinates  of  the  intersection  of  their  graphs. 

EXERCISES 


"y 

^ 

^                      *^^ 

^,             ><>      -p 

^^     2^ 

^.IZ 

^3s&. 

^^^    ^'^^ 

X'      ^1.^:1  _^-  X 

^^--u  -  ^: 

T'                                               ^ 

Y' 

254. 

equations 


Fig.  11. 

MILNE'8   IST  tr.   alo. 13 


1.    Solve  graphically  the 
Uy-Sx  =  6, 

[2x  +  Sy  =  12. 

Solution.  —  On  plotting  the  graphs 
of  both  equations,  as  in  §  248,  it  is 
found  that  they  intersect  at  a  point 
P,  whose  coordinates  are  1.8  and  2.8, 
approximately. 

Hence,    a;  =  1.8  and  y  =  2.8. 

The  coordinates  of  P  are  estimated 
to  tlie  nearest  tenth. 


194 


GRAPHIC   SOLUTIONS 


Note.  — In  solving  simultaneous  equations  by  the  graphic  method  the 
same  axes  must  be  used  for  the  graphs  of  both  equations. 

Construct  the  graphs  of  each  of  the  following  systems  of 
equations.    Solve,  if  possible.    If  there  is  no  solution,  tell  why. 


{x-y  = 


2/  =  l, 
9. 


3. 


4. 


5. 


6. 


a;  -f  2  2/  =  4. 


2x-y=z5, 
4  X  +  2/  =  16. 

3x  =  y-\-9, 
2y  =  6x-lS. 

y  =  4:X, 
x-y=S. 

I  a;  =1(^  +  4), 
[y  =  2(x-2). 


(x-\-y=  -3, 


9.     ^ 

{x-2y=-12. 

10.    I  •'^  +  2/=  4, 
[y  =  2-x. 

^^      |aj  =  2(2/  +  l), 
1  21  =  2(2  a;  -h  y). 

12.    (^+2/  =  8, 

l2x~6y=-9. 


13. 


14. 


15. 


2x  —  5y  =  5, 
10y=2x-^l. 

'Sy  =  2x-7, 
.2x  =  6  +  3y. 

'3(0^-4)  =22/, 
.6(2/ +  6)=  9a;. 


16.    |10«'  +  2/=14, 
\.Sx-5y=-2. 

^^     ^2x-\-3y  =  S, 
[3x-r2y  =  S. 


18. 


19. 


20. 


21. 


22. 


23. 


(4.y  +  3x  =  5, 
{4:X-3y  =  3. 

x  +  3y=-6, 
2x-4ry=z  -12. 

4i»-102/  =  0, 

2a;  +  2/  =  12. 

'x-2y=:2, 
^2y-Qx  =  3. 

'3a;  +  42/  =  10, 
.  6aj -f-82/  =  20. 

fa;  +  f2/  =  3i, 
10a;-22/  =  14. 


REVIEW  195 


REVIEW 


255.  1.  Distinguish  between  integral  and  fractional  equa- 
tions; between  dependent  and  independent  equations. 

2.  What  is  meant  by  the  root  of  an  equation  ?  by  solving 
an  equation  ? 

3.  Define  and  illustrate  equivalent  equations. 

4.  What  is  a  formula  ?  Give  a  simple  formula  that  has  been 
used  in  solving  some  problem. 

5.  Find  three  values  for  x  and  y  \n  x  -\- y  =  15.  What  kind 
of  an  equation  is  this,  and  why  ? 

6.  Define  simultaneous  equations ;  elimination.  State  the 
axiom  upon  which  elimination  by  addition  is  based  ;  elimination 
by  comparison. 

7.  Outline  the  method  of  elimination  by  addition  or  subtrac- 
tion; by  substitution. 

8.  State  what  is  meant  by  a  graph.  Of  what  practical  use 
are  graphs  ? 

9.  Define  abscissa ;  ordinate ;  coordinates ;  origin. 

10.  In  making  graphs,  vrhere  are  positive  values  of  x  and  y 
laid  off  ?  negative  values  ?    Interpret  the  equation  ^  =  (— 4, 3). 

11.  What  is  the  abscissa  of  any  point  of  the  y-axis?  the 
ordinate  of  any  point  of  the  ic-axis  ?  What  are  the  coordi- 
nates of  the  origin  ? 

12.  Why  are  simple  equations  sometimes  called  linear  equa- 
tions ? 

13.  Construct  the  graph  of  2?/  =  3  a;  —  4. 

14.  How  many  points  is  it  necessary  to  plot  in  drawing  the 
graph  of  a  simple  equation  ?     Why  ? 

15.  Tell  how  to  determine  where  a  graph  crosses  the  ic-axis; 
the  y-axis. 


196  REVIEW 

16.  In  drawing  the  graph  of  the  equation  3y  =  2x,  what  is 
the  result,  if  the  only  points  plotted  are  those  where  the  graph 
intersects  the  axes  ?    What  must  be  done  in  a  case  like  this  ? 

17.  Of  what  does  the  graphical  solution  of  two  simultaneous 
simple  equations  consist? 

18.  Solve  graphically  and  algebraically  : 

2x-Sy  =  10, 
5x-{-2y=    6. 
Compare  the  results  obtained. 

19.  If  a  system  of  two  linear  equations  is  indeterminate, 
how  will  this  fact  be  shown  by  the  graphs  of  the  equations 
referred  to  the  same  axes  ? 

20.  Draw  the  graphs  of  the  two  equations 
x  +  y=6y 
ic  =  13  -  ?/, 

and  tell  the  algebraic  meaning  of  the  fact  that  the  two  graphs 
do  not  intersect. 

21.  From  the  following  select  the  integral  equations;  the 
fractional  equations;  the  numerical  equations;  the  literal 
equations ;  the  indeterminate  equations : 

(1)  (3)  (6) 

Sx-\-  5y  =  19.  ax-{-bx  =  c.  5 «  +  2  =  3 a;— 10. 

(2)  (4)  (6) 

-L+l^^Sl.  2^+?^  =  31.  ^  +  ^  =  a'b^ 

2a;     3x  3  9  bx     dy 

22.  Classify  the  following  sets  of  equations  as  equivalent 
equations,  dependent  equations,  independent  equations,  simul- 
taneous equations,  or  inconsistent  equations : 

^^^   U  +  5  =  ll.  ^^^   \Sx-    y  =  2. 

(2)    p  +  .^=    7'  (4)    I    ^-    ^  =  ^' 


INVOLUTION 


256.  The  process  of  finding  any  required  power  of  an  ex- 
pression is  called  involution. 

257.  The  following  illustrate  powers  of  positive  numbers, 
of  negative  numbers,  of  powers,  of  products,  and  of  quotients, 
and  show  that  every  case  of  involution  is  an  example  of  multi- 
plication of  equal  factors. 


POWERS   OF 

A 

POWERS   OF    A 

POWERS   OF   A 

POSITIVE   NUMBER 

NEGATIVE    NUMBER 

POWER 

2  =  2^ 

-2=(-2y 

4  =  2^ 

2 

4  =  22 

-2 

4  =  (-2)2 

4 

1G  =  (22)2  =  2* 

2 

8  =  2« 

-2 

-8=(-2y 

4 

64=(22)»  =  2« 

2 
16  =  2* 

-2 

16  =  (-2)* 

4 

256  =  (22)*  =  2« 

POWER   OP   A    PRODUCT 

(2  .  3)=^  =  (2  .  3)  X  (2  .  3)  =  2  .  2  .  3  .  3  =  22 .  32. 

POWER   OP   A    QUOTIENT 

2  2^22 

3  ■  3  ~  32' 


(!)■ 


258.  From  these  examples  and  §  78,  it  is  seen  that,  for  in- 
volution : 

Law  of  Signs.  —  All  powers  of  a  positive  number  are  posi- 
tive; even  powers  of  a  negative  number  are  positive,  and  odd 
powers  are  negative. 

197 


198  INVOLUTION 

259.  From  the  examples  in  §  257  observe  that,  for  involution: 

Law  of  Exponents.  —  Tlie  exponent  of  a  power  of  a  number  is 
equal  to  the  exponent  of  the  number  multiplied  by  the  exponent  of 
the  power  to  which  the  number  is  to  be  raised. 

260.  The  last  two  examples  in  §  257  illustrate  the  following : 
Principles.  —  1.    Any  power  of  a  product  is  equal  to  the  prod- 
uct of  its  factors  each  raised  to  that  power. 

2.  Any  power  of  the  quotient  of  two  numbers  is  equal  to  the 
quotient  of  the  numbers  each  raised  to  that  power. 

261.  Axiom  6.  —  Tlie  same  powers  of  equal  numbers  are  equal. 
Thus,  if  a;  =  3,  ic2  =  32,  or  9  ;  also  x^  =  3*,  or  81 ;  etc. 

262.  Involution  of  monomials. 

EXERCISES 

1.  "What  is  the  third  power  of  4  a^6  ? 

Solution 
(4  a%)  3  =  4  a35  X  4  a^fe  X  4  a%  =  64  a^b^ 

2.  What  is  the  fifth  power  of  -  2  ab^? 

Solution 
(-2a62)5=  -2ab^x  -2ah^  x  -2ab'^x  -2ab^x  -2ab^=  -  32  aS^io. 

To  raise  an  integral  term  to  any  power : 

E,ULE.  —  liaise  the  numerical  coefficient  to  the  required,  power 
and  annex  to  it  each  letter  with  an  exponent  equal  to  the  product 
of  its  exponent  by  the  exponent  of  the  required  power. 

Make  the  power  positive  or  negative  according  to  the  law  of 
signs. 

Raise  to  the  power  indicated  : 

3.  (a6V)l  7.    (-4cy)3.  11.  (-1)^. 

4.  (a^b^cy.  8.    (-2aV)^  12.  (- 1)^^'. 

5.  (2a^cy.  9.    (abcx)"^.  13.  (3  bey. 

6.  (Ta^m'y.  10.    (2  eV)^  14.  (2  aV)«. 


•  INVOLUTION  199 


16.   What  is  the  square  of  -  ^-^9 


V      7  62c  j  It 


Solution 

6  a3x2     25  cfiv*^ 


7  h-^c  7  62c       49  6*c2 

To  raise  a  fraction  to  any  power : 

KuLE.  —  Raise  both   numerator  and   denominator  to  the   re- 
quired power  and  prefix  the  proper  sign  to  the  result. 


Raise  to  the  ■ 

power 

indicated : 

"■  & 

"  i-s- 

'"  {-■>)- 

"■  (;;)• 

"■  {-& 

"  (-&T 

-  m- 

-  (-ej- 

-  (f)-- 

263.  Involution  of  polynomials. 

The  following  are  type  forms  of  squares  of  polynomials : 

§85,  (a  +  a;)2  =  a2  +  2aa;  +  ar^. 

§  88,  (a-xy==a^-2ax-{-x'. 

§  91,  (a-x  +  yy  =  a^-{-x^-i-f-2ax-\-2ay-2xy. 

EXERCISES 

264.  Raise  to  the  second  power : 

1.  2a-^b.  4.   Sx-4.f.  7.    2«  +  35-4c. 

2.  2a-6.  5.   5m^-ll.  8.   oa^-l+^w^ 

3.  x-\-3y.  6.   4rs2  +  ^.  9.    3r'  +  2s-^t\ 

Raise  to  the  required  power  by  multiplication : 

10.  (x  +  yy.  12.    (x  +  yy.  14.    (a;  +  y)*. 

11.  (x-yy.  13.    (x-yy.  15.    (x-yy. 


200  INVOLUTION 

THE  BINOMIAL  FORMULA 

265.   By  actual  multiplication, 

(a  +  xy  =  a»  +  3  a^o;  +  3  ax^  +  a^- 

(a-xf=a'-3a'x  +  3ax''-af. 

(a-{-xy  =  a'-j-4:a^x-\-6a^x^  +  4:a3^-^x*. 

(a  —  a;)^  =  a*  —  4  a^x  -f-  6  a V  —  4  aar^  +  x\ 

From  the  expansions  just  given,  and  as  many  others  as  the 
student  may  wish  to  obtain  by  multiplication,  the  following 
observations  may  be  made  in  regard  to  any  positive  integral 
power  of  any  binomial,  the  letter  a  standing  for  the  first  term 
and  x  for  the  second : 

1.  The  7iumber  of  terms  is  one  greater  than  the  index  of  the 
required  power. 

2.  The  first  term  contains  a  only  ;  the  last  term  x  only  ;  all 
other  terms  contain  both  a  and  x. 

3.  The  exponent  of  a  in  the  first  term  is  the  same  as  the  index 
of  the  required  power  and  it  decreases  1  in  each  succeeding  term; 
the  exponent  of  x  in  the  second  term  is  1,  and  it  increases  1  in 
each  succeeding  term. 

4.  In  each  term  the  sum  of  the  exponents  of  a  and  x  is  equal 
to  the  index  of  the  required  power. 

5.  The  coefficient  of  the  first  term  is  1 ;  the  coefficient  of  the 
second  term  is  the  same  as  the  index  of  the  required  power. 

6.  The  coefficient  of  any  term  may  be  found  by  multiplying 
the  coefficient  of  the  preceding  term  by  the  exponent  of  .a  in  that 
term,  and  dividing  this  product  by  the  number  of  the  term. 

1.  All  the  terms  are  positive,  if  both  terms  of  the  binomial  are 
positive. 

8.  The  terms  are  alternately  positive  and  negative,  if  the  second 
term  of  the  binomial  is  negative. 


INVOLUTION 


201 


EXERCISES 

1.   Write  by  inspection  the  fifth  power  of  (6  ~  ?/)• 
Solution 

Substituting  b  for  a  and  y  for  x  and  applying  the  observations  of  §  265, 
(2  and  3  for  the  letters  and  exponents,  6  and  6  for  the  coefficients,  and 
8  for  the  signs)  we  have 

(b-y)^  =  b^-bb*y  +  10  b^y^  -  10  62y3  ^5  by*-  y^ 

Note.  —  Observe  that  the  coefficients  of  the  latter  half  of  the  expansion 
are  the  same  as  those  of  the  first  half  written  in  the  reverse  order. 


Expand : 

2.  {m  4-  n)^. 

3.  {m  —  nf. 

4.  (a-cy. 

5.  (a-i-bf. 

6.  (b-\-dy. 


7.  (x-yy.  12.  (a;  4- 4)". 

8.  (c-ny.  13.  {x-^5y. 

9.  (x-ay.  14.  (x-2y. 

10.  (d-yy,  15.  (a;  +  2y. 

11.  (b-\-yy.  16.  (a -3/. 

17.  Write  th«  expansion  of  (2  c^  —  5)^ 

Solution 

§  265,  (a  -  jc)4  =  a4  _  4  ^^y.  ^  q  ^2^2  _  4  ^x*  4. 3^, 

Substituting  2  c^  for  a  and  6  for  x,  we  have 

(2  c2  -  5)*  =  (2  c-^y  -  4  (2  c2)85  +  6  (2  c2)-252  -4  (2  c2)58  +  5^ 
=  16  c8  -  160  c«  +  600  c-*  -  1000  c^  +  625. 

18.  Expand  (1  -f  if^)^,  and  test  the  result. 

Solution 
§  265,  (a  +  xy  =  a^  +  Sa^x  +  Sax^-\-3^. 

Substituting  1  for  a  and  x^  for  x,  we  have 

(1  +  x2)3  =  13  +  3(l)2(a;2)  +  3(l)(x2)2  +  (a;2)8 
=  l  +  3a;2  +  3a5*  +  x«. 

Test.  —  When  a;  =  1,  (1  +  a;2)8  -  8,  and  1  +  3  2;2  +  3  x*  +  x«  =  8  ;  hence, 
(1  +  x2)«  =  1  +  3  a;2  -f  3  x*  +  x«,  and  the  expansion  is  correct. 


Expand,  and  test  results 

: 

19.    {x-\-2yy. 

23. 

(l-3a^y. 

27. 

{\-xy. 

20.    (2x-y)\ 

24. 

(5a^-a6)^. 

28. 

{\-2x)\ 

21.    {2x-b)\ 

25. 

(l  +  a262)4. 

29. 

(^-i)^-' 

22.  (x'-ioy. 

26. 

(2aa;-6)''. 

30. 

ik^-\yr- 

Expand : 

31.    (2.  +  IJ. 

34. 

("-ij- 

37. 

{t-^'l 

32.    f^-^Y. 

35. 

(-¥)■ 

38. 

{H- 

33.    (^-t\\ 

\y    ^1 

36. 

(!-¥)' 

39. 

(   ■  lA' 

40.   Expand  (r- 

-s-t)\ 

Solution 

Since  (j  —  s—  ty  may  be  written  in  the  binomial  form,  (r  —  s  —  0^, 

we  may  substitute  (r  —  s)  for  a  and  t  for  a;  in 

§  265,  (a  -  x)3  =  a3  -  3  a^x  +  3  aa:^  -  ^. 

Then,  we  have 

=  (r-  s)3  -  8(r -s)2«  +  3(r  -  s)t^-t^ 

^  |.3  _  3 ^2s  _^  3  ^s2  _  gs  _  3  ^(r2_  2  rs  +  s^)  +  3 rf^  -Sst^-  t^ 

=  ,.3  _  3  ,.2s  +  3  ys2  _  s3  _  3  ^2j  4-  6  rsi  -  3  s2^  +  3  r^2  _  3  ^^2  _  ^3. 

41.    Expand  (a  +  6  —  c  —  d)^ 


Suggestion,     (a  +  6  —  c  -  c?)^  =  (a  +  &  -  c  +  d)^,  a  binomial  form, 
Expand : 

42.  (a-\-x-yf.  46.    (a-^x  +  2y. 

43.  (a-m-n)3.  47.    {a-x-2)\ 

44.  (a-a;  +  2/)^  48.    {a  +  2h-Zc)\ 

45.  (a-x-yy.  49.    (a  +  &  +  a^  +  2/)'- 


EVOLUTION 


267.  Just  as  (§  132)  one  of  the  two  equal  factors  of  a  number 
is  its  second,  or  square,  root ;  so  one  of  the  three  equal  factors  of 
a  number  is  its  third,  or  cube,  root;  one  of  the  four  equal  factors, 
the  fourth  root ;  etc. 

The  second  root  of  a  number,  as  a,  is  indicated  by  Va ;  the 
third  root  by  \/a ;  the  fourth  root  by  ■\/a ;  the  fifth  root  by 
Va;  etc. 

The  sign  ^  is  called  the  root  sign,  or  the  radical  sign ;  the 
small  figure  in  its  opening  is  called  the  index  of  the  root. 

When  no  index  is  written,  the  second,  or  square,  root  is 
meant. 

268.  The  process  2«  =  2  •  2  •  2  =  8  illustrates  involution. 
The  process  v^8  =  ^2  •  2  •  2  =  2  illustrates  evolution,  which 

will  be  defined  here  as  the  process  of  finding  a  root  of  a  num- 
ber, or  as  the  inverse  of  involution, 

269.  You  have  learned  (§  132)  that  every  number  has  two 
square  roots,  one  positive  and  the  other  negative. 

For  example,  V25  =  +  5  or  —  5. 

The  roots  may  be  written  together,  thus :  ±5,  read  ^plus  or 
minus  Jive  \'  or  t5,  read  ^  minus  or  plus  Jive,^ 

270.  The  square  root  of  — 16  is  not  4,  for  4^  =  -f  16 ;  nor 
—  4,  for  (—4)^  =  +  16.  No  number  so  far  included  in  our 
number  system  can  be  a  square  root  of  —  16  or  of  any  other 
negative  number. 

It  would  be  inconvenient  and  confusing  to  regard  Va  as  a 
number  only  when  a  is  positive.  In  order  to  preserve  the 
generality  of  the  discussion  of  number,  it  is  necessary,  there- 


204  EVOLUTION 

fore,  to  admit  square  roots  of  negative  numbers  into  our  num- 
ber system.     The  square  roots  of  — 16  are  written 

V^=36and  -V^=n[6. 

Such  numbers  are  called  imaginary  numbers  and,  in  contrast, 
numbers  that  do  not  involve  a  square  root  of  a  negative  num- 
ber are  called  real  numbers. 

271.  Just  as  every  number  has  two  square  roots,  so  every 
number  has  three  cube  roots,  four  fourth  roots,  etc. 

For  example,  the  cube  roots  of  8  are  the  roots  of  the  equa- 
tion x^  =  8,  which  later  will  be  found  to  be 

2,-1+  \^^^,  and  -  1  -  V^^ 
The  present  discussion  is  concerned  only  with  real  roots. 

272.  Since  2=^  =  8,  ^8  =  2. 
Since  {-2y  =  -S,  ,      ^ir8  =  _2.  • 
Since            2^  =  16  and  (  -  2)^  =  16,            ^16  =  ±  2. 
Since           2^^  =  32,                                         V^=2. 
Since    (-2)«=-32,                                 ■^/'=^  =  -2. 

A  root  is  odd  or  even  according  as  its  index  is  odd  or  even. 

273.  It  follows  from  the  preceding  illustrations  and  from 
the  law  of  signs  for  involution  (§  258)  that,  for  real  roots : 

Law  of  Signs.  —  An  odd  root  of  a  number  has  the  same  sign  as 
the  number. 

An  even  root  of  a  iiumber  may  have  either  sign. 

274.  A  real  root  of  a  number,  if  it  has  the  same  sign  as  the 
number  itself,  is  called  a  principal  root  of  the  number. 

The  principal  square  root  of  25  is  5,  but  not  —  5.  The  principal  cube 
root  of  8  is  2  ;  of  -  8  is  -  2. 

275.  Axiom  7.  —  The  same  roots  of  equal  numbers  are  equal. 
Thus,  if  «  =  16,  Vx  =  4  ;  if  a:  =  8,  v^  =  2  ;  etc. 


EVOLUTION  205 

276.  Since  (2^  =  2^""^  =  2«,  the  principal  cube  root  of  2«  is 

Hence,  for  evolution : 

Law  of  Exponents.  —  The  exponent  of  any  root  of  a  number  is 
equal  to  the  exponent  of  the  number  divided  by  the  index  of  the 
root.  • 

277.  Since  (5  a)^  =  ^^d^  =  25  a^,  the  principal  square  root  of 
25  a"  is  _ 

V25^  =  V25.  -/^=5a. 

Hence,  for  principal  roots  : 

Principle.  —  Any  root  of  a  product  may  be  obtained  by  takirig 
that  root  of  each  of  the  factors  and  finding  the  product  of  the 
results. 

278.  Since  f  -  ]  =  ^  =  ^,  the  principal  fourth  root  of  —  is 

\Sj      3*     81'        ^        ^  81 

^81      ^81      </S'     3* 
Hence,  for  principal  roots : 

Principle.  —  Any  root  of  the  quotient  of  two  numbers  is  equal 
to  that  root  of  the  dividend  divided  by  that  root  of  the  divisor. 

279.  Evolution  of  monomials. 

exercises 
1.   Find  the  square  root  of  36  a«6l 

Solution 

Since,  in  squaring  a  monomial,  §  262,  the  coefficient  is  squared  and  the 
exponents  of  the  letters  are  multiplied  by  2,  to  find  the  square  root,  the 
square  root  of  the  coefficient  must  be  found,  and  to  it  must  be  annexed 
the  letters  each  with  its  exponent  divided  by  2. 

The  square  root  of  36  is  6,  and  the  square  root  of  the  literal  factors  is 
a^h.    Therefore,  the  principal  square  root  of  36  a^h'^  is  6  a^h. 

The  square  root  may  also  be  —  6  a%.,  since  —Qa^h  x  —  6  a^6  =  36  a%K 


.'.  VS6 a662  =  ±  6 a«ft. 


206  EVOLUTION 

2.   Find  the  cube  root  of  —  125  x^y^\ 
Solution 


V  —  125  a^j/'-^i  =  —  5  x'^y'^,  the  real  root. 

To  find  the  root  of  an  integral  term : 

Rule.  —  Find  the  required  root  of  the  numerical  coefficient, 
annex  to  it  the  letters  each  with  its  exponent  divided  by  the  index 
of  the  root  sought,  and  prefix  the  proper  sign  to  the  result. 

Find  real  roots : 


4.  Va¥V^  9.  V-32a;iYo.  14.  ^  (_  ^25)9. 

5.  ^/aV/".  10-  Vl6^y.  15-  --v/^is&V^. 

6.  ^a*"6Vl  11-  -v/-a^i535^i^  16.  -  ^_27/)V. 

7.  V^V^  ^^-  a/  -  243  /".  17.  -A/'-128ai%28. 

18.   Find  the  cube  root  of  27  m^n^^' 

Solution 

»/_8a;9y6  ^  v/-8xV  ^  -  2  x^y^  ^  _  2x^ 

To  find  the  root  of  a  fractional  te'rm : 

Rule.  —  Find  the  required  root  of  both  numerator  and  denom- 
inator, and  prefix  the  proper  sign  to  the  resulting  fraction. 

Find  real  roots  : 


19.    J^^^y.        21.   x7-125^.  23.    4 

■\81mV  ^  y^  ^ 


64  6« 


20.    #-y)"       22.   .^/^32aVo.       ^^^     s/_  125^ 
\l28a;"  \    243  y^^  \       1728  c^ 


EVOLUTION  207 

280-  To  find  the  square  root  of  a  polynomial. 

EXERCISES 

1.   Derive    the  "  process    for    finding    the    square    root    of 

PROCESS 

a^^2ab  +  b^\a±b 

^ 

Trial  divisor,         2  a 


Complete  divisor,  2  a  +  b 


2ab-{-b^ 
2ab-\-b^ 


Explanation.  —  Since  a^  -f-  2  a6  +  6^  is  the  square  of  (a  +  6),  we  know 
that  the  square  root  of  a^-\- 2  ab  +  b^  is  a +  b. 

Since  the  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the 
square  root  of  a^,  the  first  term  of  the  power.  On  subtracting  a^,  there 
is  a  remainder  of  2  a&  +  b^. 

The  second  term  of  the  root  is  known  to  be  6,  and  that  may  be  found 
by  dividing  the  first  term  of  the  remainder  by  twice  the  part  of  the  root 
already  found.     This  divisor  is  called  a  trial  divisor. 

Since  2  a6  +  6^  is  equal  to  6  (2  a +  6),  the  complete  divisor  which 
multiplied  by  b  produces  the  remainder  2  ab -^b^  \s2a  +  b  ;  that  is,  the 
complete  divisor  is  found  by  adding  the  second  term  of  the  root  to  twice 
the  root  already  found. 

On  multiplying  the  complete  divisor  by  the  second  term  of  the  root 
and  subtracting,  there  is  no  remainder  ;  then,  a  +  6  is  the  required  root. 

2.   Find  the  square  root  of  9  a:^  _  3Q  ^^  _f_  25  y2 

PROCESS 

9 sp^ -SO xy -^25  f\S  X - 5  y 
9«2 

Trial  divisor,  6  x 


30  0^2/ +  25/ 
3Qa;?/  +  25/ 


Complete  divisor,  6x  —  5y 
Find  the  square  root  of : 

3.  4:X^-{.12x-^9,  6.   c2-12c4-36. 

4.  ar^  +  2a;  +  l.  7.   4ar^  +  4ic  +  l. 

5.  l-4m+4ml  8.   16 -{-24.x +  9 a^. 

Since,  in  squaring  a  +  b-\-c,  a  +  b  may  be  represented  by  x, 
and  the  square  of  the  number  by  a^  +  2  xc  4-  c^,  the  square  root 


208  EVOLUTION 

of  a  number  whose  root  consists  of  more  than  the  two  terms  may 
be  obtained  in  the  same  way  as  in  exercise  1,  by  considering  the 
terms  already  found  as  one  term. 

9.    Find  the  square  root  of  4  a^^  +  12  ar^  -  3  a;^  -18  a;  +  9. 

PROCESS 


4  a;^  +  12  a^  -  3  a!- -  18  a;  +  9|2  a;- +  3  a; - 
4a;^ 

-3 

4a^ 

4  a?2  +  3  a; 

12  ar*  -  3  x' 
12  aj'  +  9  x' 

4.x'-{-(5x 
4:x^  -i-6x 

-3 

-12a;2-18aj  +  9 
-  12  a.-^- 18a; +  9 

Explanation. — Proceeding  as  in  exercise  2,  we  find  that  the  first  two 
terms  of  the  root  are  2a:2  +  Sx. 

Considering  (2  x^  +  3  x)  as  the  first  term  of  the  root,  we  find  the  next 
term  of  the  root  as  we  found  the  second  term,  by  dividing  the  remainder 
by  twice  the  part  of  the  root  already  found.  Hence,  the  trial  divisor  is 
4  a:^  +  6  X,  and  the  next  term  of  the  root  is  —  3.  Annexing  this,  as  before, 
to  the  trial  divisor  already  found,  we  find  that  the.complete  divisor  is 
4  x2  +  6  X  —  3.  Multiplying  this  by  —  3  and  subtracting  the  product  from 
—  12  a;2  -  18  x  +  9,  we  have  no  remainder. 

Hence,  the  square  root  of  the  number  is2  x'^  +  S  x  —  S. 

Rule.  —  Arrange  the  terms  of  the  polynomial  with  reference  to 
the  consecutive  powers  of  some  letter. 

Find  the  square  root  of  the  first  term,  write  the  result  as  the  first 
term  of  the  root,  and  subtract  its  square  from  the  given  polynomial. 

Divide  the  first  term  of  the  remainder  by  twice  the  root  already 
found,  used  as  a  trial  divisor,  and  the  quotient  will  be  the  next 
tei'm  of  the  root.  Write  this  result  in  the  root,  and  annex  it  to  the 
trial  divisor  to  form  the  complete  divisor. 

Midtiply  the  complete  divisor  by  this  term  of  the  root,  and  sub- 
tract the  product  from  the  first  remainder. 

Find  the  next  term  of  the  root  by  dividing  the  first  term  of  the 
remainder  by  the  first  term  of  the  trial  divisor. 

Form  the  complete  divisor  as  before  and  continue  in  this  man- 
ner until  all  the  terms  of  the  root  have  been  found. 


EVOLUTION  209 

Find  the  square  root  of : 

10.  25a2-40a  +  16.  12.    ar^  +  rcy+Jyl 

11.  900a^  +  60a;  +  l.  13.    4  a;^  -  52  x^  + 169. 

14  9  a*  -  12  ar*  4- 10  x-  -  4  a;  +  1 . 

16  x'-6xhj-^13a^-12xf-\-4p*. 

16.  a.-«  +  2aV-aV-2aV  +  al 

17.  25a;*H-4-12a;-30a:3_^29a^. 

18.  l-2x-\-Sa^-4:X^  +  Sx'-2x'-\-a^. 

19.  |_4^V4a^  +  f -2a6  +  |^ 
20.   Find  four  terms  of  the  square  root  of  1  +  a;. 

SQUARE  ROOT  OF   ARITHMETICAL  NUMBERS 

281.  Compare  the  number  of  digits  in  the  square  root  of 
each  of  the  following  numbers  with  the  number  of  digits  in 
the  number  itself : 


NUMBER 

ROOT 

NUMBER 

ROOT 

NUMBER 

ROOT 

1 

1 

I'OO 

10 

I'OO'OO 

100 

25 

6 

10'24 

32 

56'25'00 

750 

81 

9 

98'01 

99 

99'80'01 

999 

From  the  preceding  comparison  it  may  be  observed  that : 

Principle.  —  If  a  number  is  separated  into  periods  of  two 
digits  each,  beginning  at  units,  its  square  root  will  have  as  many 
digits  as  the  number  has  periods. 

The  left-hand  period  may  be  incomplete,  consisting  of  only  one  digit. 

282.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  t  and  the  number  of  units  expressed  by  the 
units'  digit  by  ?*,  any  number  consisting  of  tens  and  units  will 
be  represented  by  t-\-u,  and  its  square  by  {t-\-u)'\  or  ^^+2  tu->ru\ 

Since  25  =  20  +  5,  252  _  ^20  +  5)2  =  20^  +  2(20  x  6)  +  62  =  626. 

MILNE'S    IST    YR.    A  !.0    — 14 


2^=120 
w=     2 


2^  +  ^  =  122 


210  EVOLUTION 

EXERCISES 

283.    1.    Find  the  square  root  of  3844. 

FIRST    PROCESS 

38'44l60-l-2  Explanation.  —  Separating    the 

I — —       number  into  periods  of  two  digits 

^^^^  each  (Prin.,  §  281),  we   find  that 

2  44  the  root  is  composed  of  two  digits, 

tens  and  units.      Since  the  largest 
2  44  square  in  38  is  6,  the  tens  of  the  root 

cannot  be  greater  than  6  tens,  or  60. 
Writing  6  tens  in  the  root,  squaring,  and  subtracting  from  3844,  we  have 
a  remainder  of  244. 

Since  the  square  of  a  number  composed  of  tens  and  units  is  equal  to 
{the  square  of  the  tens)  +  {twice  the  product  of  the  tens  and  the  units)  + 
(the  square  of  the  units).,  when  the  square  of  the  tens  has  been  subtracted, 
the  remainder,  244,  is  twice  the  product  of  the  tens  and  the  units,  plus 
the  square  of  the  units,  or  only  a  little  more  than  twice  the  product  of  the 
tens  and  the  units. 

Therefore,  244  divided  by  twice  the  tens  is  approximately  equal  to  the 
units.  2x6  tens,  or  120,  then,  is  a  trial,  or  partial  divisor.  On  dividing 
244  by  the  trial  divisor,  the  units'  figure  is  found  to  be  2. 

Since  twice  the  tens  are  to  be  multiplied  by  the  units,  and  the  units 
also  are  to  be  multiplied  by  the  units  to  obtain  the  square  of  the  units,  in 
order  to  abridge  the  process  the  tens  and  units  are  first  added,  forming 
the  complete  divisor  122,  which  is  then  multiplied  by  the  units.  Thus, 
(120  +  2)  multiplied  by  2  =  244. 

Therefore,  the  square  root  of  3844  is  62. 

SECOND    PROCESS 


Explanation.  —  In  practice  it  is  usual 
to  place  the  figures  of  the  same  order  in 
the  same  column,  and  to  disregard  the 
ciphers  on  the  right  of  the  products. 


Since  any  number  may  be  regarded  as  composed  of  tens  and 
units,  the  foregoing  processes  have  a  general  application. 

Thus,  346  =  34  tens  +  6  units  ;  2377  =  237  tens  +  7  units. 


t'= 

38'44[62 
36 

2^=120 
u=     2 

2  44 

2t-^u  =  122 

244 

EVOLUTION  211 

2.  Find  the  square  root  of  104976. 
Solution 


10'49'76| 
9 


Trial  divisor  =  2  x    30  =  60 

Complete  divisor  =  60  +    2  =62 


1  49 
1  24 


Trial  divisor  ==  2  x  320  =  640 

Complete  divisor  =  640  +  4  =  644 


25  7€ 
25  76 


Rule.  —  Separate  the  number  into  periods  of  two  figures  each, 
beginning  at  units. 

Find  the  greatest  square  in  the  left-hand  period  and  write  its 
root  for  the  first  figure  of  the  required  root. 

Square  this  root,  subtract  ike  result  from  the  left-hand  period, 
and  annex  to  the  remainder  the  next  period  for  a  new  divi- 
dend. 

Double  the  root  already  found,  with  a  cipher  annexed,  for  a 
trial  divisor,  and  by  it  divide  the  dividend.  The  quotient,  or 
quotient  diminished,  will  be  the  second  figure  of  the  root.  Add  to 
the  tried  divisor  the  figure  last  found,  multiply  this  complete  divisor 
by  the  figure  of  the  root  last  found,  subtract  the  product  from  the 
dividend,  and  to  the  remainder  annex  the  next  period  for  the  next 
dividend. 

Proceed  in  this  manner  until  all  the  periods  have  been  used. 
The  result  will  be  the  square  root  sought. 

1.  When  the  number  is  not  a  perfect  square,  annex  periods  of  decimal 
ciphers  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  tovs^ard  the  right. 

3.  The  square  root  of  a  common  fraction  may  be  obtained  by  finding 
the  square  root  of  both  numerator  and  denominator  separately  or  by 
reducing  the  fraction  to  a  decimal  and  then  finding  the  root. 

Find  the  square  root  of: 

3.  529.                     6.  57121.  9.  2480.04. 

4.  2209.                     7.  42025.  10.  10.9561. 

5.  4761.                     8.  95481.  11.  .001225. 


212  EVOLUTION 

Find  the  square  root  of : 

12.  m.      14.  if|.       16.  ifj.      18.  m 

13.    lif.  15.    tV¥^.  17.    lU-  19.    Iff. 

Find  the  square  root  to  two  decimal  places : 

20.  |.  22.    |.  24.    f.  26.    J. 

21.  |.  23.    .6.  25.    |.  27.    yV 

ROOTS  BY  FACTORING 

284.  The  method  of  finding  the  cube  root  of  polynomials 
and  of  arithmetical  numbers,  analogous  to  the  one  just  given 
for  square  root,  is  beyond  the  scope  of  this  text ;  but  a  method 
of  finding  the  cube  root,  or  any  other  root,  of  a  number  that  is 
a  perfect  power  of  the  same  degree  as  the  index  of  the  required 
root  is  here  mentioned  because  of  its  simplicity. 

This  method  consists  in  factoring,  grouping  the  factors,  and 
taking  the  required  root  of  each  group. 


Thus, 

V42875  =  ^5  .  5  .  5  X  7  .  7  .  7  =  ^63  X  73  =  5  X  7  z 

also, 

Va:*  +  2  x3  _  3  a;2  -  4  a;  +  4  =  V(x  -  l)\x  +  2)2 

=  (x-l)(x-h2) 

• 

=  x^^x-2. 

EXERCISES 

285.    Find,  by  the  method  of  factoring  : 

1.  Square  root  of  a«  -  12  a^  +  36. 

2.  Guhevootoix^-15a:P-\-75x-125. 

3.  Fourth  root  of  a;^  -  8  a.-«  +  24  x^  -  32  a;  +  16. 

4.  Fifth  root  of  a^  -  10  ic*  +  40  a^  -  80  a^  -f  80  a;  -  32. 
Find  the  indicated  root : 

5.  ^3375.  7.    -v/262144.  9.    ^/4084101. 

6.  ^1296.  8.    V759375.  10.    ■v/16777216. 


RADICALS 


286.  Thus  far  the  exponents  used  have  been  positive  integers 
only,  and  the  laws  of  exponents  have  been  based  on  this  idea ; 
but  since  zero,  fractional,  and  negative  exponents  may  occur  in 
algebraic  processes,  they  must  follow  the  same  laws  as  are 
given  for  positive  integral  exponents ;  hence,  it  becomes  neces- 
sary to  discover  meanings  for  these  new  kinds  of  exponents, 
because,  for  example,  in  aP,  a~^,  and  at,  the  exponents  0,-2,  and 
•J  cannot  show  how  many  times  a  is  used  as  a  factor  (§9). 

287.  Meaning  of  zero  and  negative  exponents. 

By  notation,  §§  9,  10,  a^  =  1  .  a  •  a.  (1) 

Dividing  both  members  of  this  equation  by  a,  the  first  mem- 
ber by  subtracting  exponents  (§  32)  and  the  second  by  taking 
out  the  factor  a,  we  have 

a}  =  l'a.  (2) 

Dividing  (2)  by  a,  a«  =  1.  (3) 

Dividing  (3)  by  a,  o"^  =  -•  (4) 

Dividing  (4)  by  a,  a"^  =  — •  (5) 

The  meaning  of  a  zero  exponent,  illustrated  in  (3),  and  of  a 
negative  exponent,  in  (4)  and  (5),  may  be  stated  as  follows : . 

Any  number  with  a  zero  exponent  is  equal  tol. 

Any  number  with  a  negative  exponent  is  equal  to  the  reciprocal 
of  the  same  number  with  a  numerically  equal  positive  exponent. 

213 


214  RADICALS 

288.  The  meaniDg  of  a  negative  exponent  shows  that: 

Principle.  —  Any  factor  may  he  transferred  from  one  term  of 
a  fraction  to  the  other  without  changing  the  value  of  the  fraction, 
provided  the  sign  of  the  exponent  is  changed. 

289.  Meaning  of  a  fractional  exponent. 
Just  as,  §  276,         Vo^  =  a^-^  =  a, 

so  -y/o?  =  a^^^  =  ai.     That  is. 

The  numerator  of  a  fractional  exponent  with  positive  integral 
terms  indicates  a  power  and  the  denominator  a  root. 

Since  the  operations  may  be  performed  in  either  order : 
The  fractional  exponent  as  a  whole  indicates  a  root  of  a  power 
or  a  power  of  a  root, 

EXERCISES 

290.  Find  a  simple  value  for : 

1.  5«.  3.    2-\  5.    (-3)«.  7.    (a%\y. 

2.  4-1  4.    3-1  6.    (-6)-l  8.    (-J)"'. 
9.   Which  is  the  greater,  (4.)^  or  {\Y?  (i)-^  or  (^)-3? 

10.  Write  5  aj~y  with  positive  exponents. 
Solution.  —  By  §  287 ,     5  x-^y'^  =  by'^—  =  ^. 

Write  with  positive  exponents  : 

11.  2a;-\  13.    a'^^h-K  15.   4^ah-\ 

12.  5a-\  14.    x-^y-^.  16.   Saa;-^ 

3  a^v 
17.    Write  — f-  without  a  denominator. 
hoir 

Solution.  —  By  §  288,        ^^  =  3  a^ft-^-ay. 


Write  without  a  denominator 
ax  _     mn 

by  '    a^  ""   a~^lf 


18.    ^.  19.    ???.  20.    -U-  21.    "''" 


RADICALS  215 

22.  Find  the  value  of  16^. 

First  Solution.     16^  =  ^/W  =  v  16  •  16  •  16 

=  v''(2  .  2  .  2  .  2)  (2  .  2  .  2  •  2)  (2  .  2  .  2  .2  ) 
=  v/(2  .  2  •  2) (2  .  2  .  2) (2  .  2  .  2)(2  . 2  . 2) 
=  2.2-2  =  8. 

Second  Solution.     16*  =  (16^)3  =  23  =  8. 

In  numerical  exercises  it  is  usually  best  to  find  the  root  first. 

Simplify,  taking  only  principal  roots : 

23.  Sl  25.    64i  27.    64"^. 

24.  si  26.    32^  28.    (-8)"! 
29.   Which  is  the  greater,  27^  or  (-  27)"^?  (})^  or  (i)~^? 


30.  Express  va^6c'~'*  with  positive  fractional  exponents. 

2     1 

Solution.  y/a^bc^  =  ahh~^  =  ^-^ . 

Express  with  positive  fractional  exponents : 

31.  Va63.  33.    (Vxf,  35.    (\/^)-l 

32.  ^xy.  34.    (a/?/)*.  36.    SVaJ-^"*- 

Express  roots  with  radical  signs  and  powers  with  positive 
exponents : 


37.  ^\ 

39. 

xK 

41. 

x^y^. 

43.  a^-^x^. 

38.    xi 

40. 

ahi 

42. 

ah-y 

44.  x^^2/^. 

Multiply : 

45.   a^  by  a-\ 

47. 

a*  by  a-*. 

49. 

a^  by  a*'. 

46.   a^bya-^ 

48. 

a  by  a-\ 

50. 

a;i  by  xK 

Divide : 

51.   a«bya«. 

53. 

a^  by  a -I 

55. 

a;  2  by  jci 

52.   a^  by  a®. 

54. 

ar^  by  «"  2. 

56. 

a;""^  by  a^-l 

216  RADICALS 

Solve  for  values  of  x  corresponding  to  principal  roots  by 
applying  axioms  6  and  7  (§§  261,  275),  and  test  each  result; 

57.  x^  =  l.  .  61.  a;~^  =  6. 

58.  aj^  =  8.  62.  a;"t  =  144. 

59.  a;^  =  81.  63.  25a;-^  =  l. 

60.  \x^=12.  64.  a;^  +  32  =  0. 

291.  An  indicated  root  of  a  number  is  called  a  radical;  the 
number  whose  root  is  required  is  called  the  radicand. 

\/5a,  (a^)^,  y/d^  +  2,  and  {x-\-  y)*  are  radicals  whose  radicands  are, 
respectively,  5  a,  x^,  a^  +  2,  and  x-{-y. 

292.  The  order  of  a  radical  is  shown  by  the  index  of  the 
root  or  by  the  denominator  of  the  fractional  exponent. 

y/a  +  x.and  (6  —  xy  are  radicals  of  the  second  order. 

293.  In  the  discussion  and  treatment  of  radicals  only  pritv- 
cipal  roots  will  be  considered. 

Thus,  Vl6  will  be  taken  to  represent  only  the  principal  square  root  of 
16,  or  4.     The  other  square  root  will  be  denoted  by  —  VlG. 

294.  Graphical  representation  of  a  radical  of  the  second  order. 
In  geometry  it   is  shown  that   the   hypotenuse  of   a  right 

triangle  is  equal  to  the  square  root  of  the  sum  of  the  squares  of 
the  other  two  sides;  consequently,  a  radical  of  the  second  order 
may  be  represented  graphically  by  the  hypotenuse  of  a  right 
triangle  whose  other  two  sides  are  such  that  the  sum  of  their 
squares  is  equal  to  the  radicand. 

Thus,  to  represent  V5  graphically,  since  it  may  be  observed 
that  5  =  22  + 12,  draw  OA  2  units  in 
length,  then  draw  AB  1  unit  in  length 
in  a  direction  perpendicular  to  OA. 
Draw  OB,  completing  the  right-angled 
triangle  OAB.  Then,  the  length  of  OB 
represents  V5  in  its  relation  to  the  unit  length. 


RADICALS  217 

EXERCISES 

295.  Eepresent  graphically : 

1.  V2.  3.    Vl3.  5.    V34.  7.    V|. 

2.  ViO.  4.    Vl7.  6.    V25.  8.    Vif. 

296.  A  number  that  is,  or  may  be,  expressed  as  an  integer 
or  as  a  fraction  with  integral  terms,  is  called  a  rational  num- 
ber. 

3.  I,  V^,  and  .333  are  rational  numbers. 

297.  A  number  that  cannot  be  expressed  as  an  integer  or  as 
a  fraction  with  integral  terms  is  called  an  irrational  number. 

v^^  4^^  1  _^  y/S^  and  V  1  +  V3  are  irrational  numbers. 

From  §  294,  it  will  be  observed  that  the  irrational  number  V5  can  be 
represented  graphically  by  a  line  of  exact  length,  though  it  cannot  be 
represented  exactly  by  decimal  figures,  for  Vb  =  2.236...,  which  is  an  end- 
less decimal. 

298.  When  the  indicated  root  of  a  rational  number  cannot 
be  obtained  exactl}'',  the  expression  is  called  a  surd. 

V2  is  a  surd,  since  2  is  rational  but  has  no  rational  square  root. 
V  1  +  V3  is  not  a  surd,  because  1  +  V3  is  not  rational. 
Radicals  may  be  either  rational  or  irrational,  but  surds  are 
always  irrational. 

Both  Vl  and  \/S  are  radicals,  but  only  V3  is  a  surd. 

299.  A  surd  may  contain  a  rational  factor,  that  is,  a  factor 
whose  radicand  is  a  perfect  power  of  a  degree  corresponding 
to  the  order  of  the  surd. 

The  rational  factor  may  be  removed  and  written  as  the  co- 
efficient of  the  irrational  factor. 

In  \/8  =  V'4  X  2  and  Vbi=  \/27  x  2,  the  rational  factors  are  Vi  and 
\/27,  respectively ;  that  is,  VS  =  2  v^  and  V54  =  3  y/2. 

300.  In  the  following  pages  it  will  he  assumed  that  irrational  num- 
bers obey  the  same  law  as  rational  numbers.  For  proofs  of  the  generality 
of  these  laws,  the  reader  is  referred  to  the  author's  Advanced  Algebra. 


218  RADICALS 


REDUCTION  OF  RADICALS 

301.   To  reduce  a  radical  to  its  simplest  form. 

As  the  work  progresses  the  student  will  discover  the  mean- 
ing of  simplest  form. 

EXERCISES 


302.    1.   Eeduce  V20a^to  its  simplest  form  by  writing  the 
rational  factor  as  the  coefficient  of  the  irrational  factor. 


PROCESS 


V20a«=  V4a«x5  =  V4a«x  V5=2aV5 

Explanation.  —  Since  the  highest  factor  of  20  a^  that  is  a  perfect  square 
is  4  a^,  \/20  a^  is  separated  into  two  factors,  a  rational  factor  \/4  a^,  and 
an  irrational  factor  VS  ;  that  is,  §  277,  V'20  a^  =  ■\/4  a'^  x  \/5. 

On  finding  the  square  root  of  4  a^  and  prefixing  the  root  to  the  irrational 
factor  as  a  coefficient,  the  result  is  2  a^Vb. 


2.   Reduce  V  —  864  to  its  simplest  form. 


PROCESS 

3/ KTTi  3/ 


</_  864  =  V- 216  X  4  =  V  -  216  X  V4  =  -  6S/4 

EuLE.  —  Separate  the  radical  into  two  factors  one  of  which  is 
its  highest  rational  factor. 

Find  the  required  root  of  the  rational  factor,  multiply  the  result 
by  the  coefficient,  if  any,  of  the  given  radical,  and  place  the  product 
as  the  coefficient  of  the  irrational  factor. 

Eeduce  to  simplest  form : 

3.    Vl2.  8.    -^32.  13.    V243  aV^. 


4. 

V75. 

9. 

V18  a\ 

14. 

-v/128  a%\ 

5. 

<m. 

10. 

V25  6. 

15. 

(a^-\-5a')l 

6. 

V128. 
-\/250. 

11. 
12 

V98  c«. 
VBOa. 

16. 
17. 

Vl8a;-9. 

7. 

</^-2at. 

RADICALS  219 

18.  Reduce  \.y-^  to  its  simplest  form ;  that  is,  to  a  radical 
having  an  integral  radicand. 

PROCESS 

Explanation.  —  Since  the  denominator  must  be  removed  from  the 
radical,  and  since  the  radical  is  of  the  second  order, -the  denominator  must 
be  made  a  perfect  square.     The  smallest  factor  that  will  do  this  is  2  y. 

On  multiplying  both  terms  of  the  fraction  by  this  factor,  the  largest 

rational  factor  of  the  resulting  radical  is  found  to  be  A,'-^,  or  -^. 
Therefore,  the  irrational  factor  is  ^^^2  y  and  its  coeflBcient  is  - — ;. 

^yi- 

Reduce  to  simplest  form  : 

19-    V|.  24.    ^/2?.  27.    SL 

20.    V|.  ^_^  ^-^^ 


21. 

22.   -\/4.  r  ro — 

^  26.       ^'^  ****         \  ox 


23.    Vt%.  ^y  \50aV 

303.  Although  |  =  J,  it  does  not  follow  that  64t  =  64',  for 
each  fractional  exponent  denotes  a  power  of  a  root  of  64,  and 
the  roots  and  powers  taken  are  not  the  same  for  64^  as  for  64t. 
By  trial,  however,  it  is  found  that  each  number  is  equal  to  8 ; 
and  in  general  it  may  be  proved  that 

A  number  having  a  fractional  exponent  is  not  changed  in  value 
by  reducing  the  fractional  exponent  to  higher  or  lower  terms. 

EXERCISES 

304.  1.  Reduce  V9a^  to  its  simplest  form ;  that  is,  to  a 
radical  having  the  smallest  index  possible. 

PROCESS 

3  —  rir,\^ 


79  a'  =  ^/(S  af  =  (3  ay  =  (3  ay  =  V3  a 


220  RADICALS 


2.  Reduce  v64a^to  its  simplest  form. 

PROCESS 

■v/64a«6^=  v/2W¥  =  6(2a6)^  =  6(2a&)*=  6^4a¥ 
Simplify : 

3.  ^36.  6.    ^1600.  7.    ^9  a'Wc\ 

4.  -V2b.  .        6.    \/27^.  8.    a/121  aV. 

305.  The  student  has  doubtless  discovered  that : 

A  radical  is  in  its  simplest  form  when  the  index  of  the  root 
is  as  small  as  possible,  and  when  the  radicand  is  integral  and 
contains  no  factor  that  is  a  perfect  power  whose  exponent 
corresponds  with  the  index  of  the  root. 

V?  is  in  its  simplest  form  ;  but  \/|  is  not  in  its  simplest  form,  because 
I  is  not  integral  in  form  ;  VS  is  not  in  its  simplest  form,  because  4,  a 

6/ 1 

factor  of  8,  has  an  exact  square  root ;   v25,  or  25^,  is  not  in  its  simplest 
form,  because  25^  =  (5^)^  =  5^  =  5^  or  v^5. 

MISCELLANEOUS  EXERCISES 

306.  Eeduce  to  simplest  form-: 

1.  V600.  5.    a/189.  9.    </li4.  13.    VJ. 

2.  V500.  6.    VM.  10.    \/8i. 


14. 


4.    A^MOO.        8.    \/l92.  12.    v/289.  15.    ^/^ 


3.    ^160.  7.    a/72.  11.    \/343.  ^ 


16.  V405  ahf.  18.    VS  -  20  61  20.    Va'h%^d\ 

17.  (135  «y)^.  19.   5V4a2  +  4.  21.    (16  a;  -  16)^. 

307.    A  surd  that  has  a  rational  coefficient  is  called  a  mixed 
surd. 

2'n/2,  a  v^,  and  (a  -  6)  Va  +  6  are  mixed  surds. 


RADICALS  221 

308.  A  surd  that  has  no  rational  coefficient  except  unity  is 
called  an  entire  surd. 

V5,  \/lT,  and  VoM^  are  entire  surds. 

309.  To  reduce  a  mixed  surd  to  an  entire  surd. 

EXERCISES 

1.   Express  2  aVS^  as  an  entire  surd. 

PROCESS 


2  a  V5  6  =  V4  aV5  h  =  V4a^x56  =  V20  a'^6 

Rule.  —  Raise  the  coefficient  to  a  power  corresponding  to  the 
index  of  the  given  radical,  and  introduce  the  result  under  the 
radical  sign  as  a  factor. 

Express  as  entire  surds : 

2.  2V2.  5.   3-5/3.  8.   |V2.  11.    \V^. 

3.  3V5.  6.   4V5.  9.    |V^\  12.   fV'lf^^. 

4.  5V2.  7.   i^/^.  10.    |V6c.         13.    1^. 
310.   To  reduce  radicals  to  the  same  order. 

EXERCISES 

1.   Reduce  \/3,  V2,  and  V4  to  radicals  of  the  same  order. 

PROCESS 

■^3  =  3*  =  3A  =  ^=^27 

V2  =  22  =  2^=^2^=  ^^64 

Rule.  —  Eocpress  the  given  radicals  with  fractional  exponents 
having  a  common  denominator. 

Raise  each  number  to  the  power  indicated  by  the  numerator  of 
its  fractional  exponent,  and  indicate  the  root  expressed  by  the 
common  denominator. 


222  RADICALS 

Reduce  to  radicals  of  the  same  order : 

2.  V2  and  </3.  7.   v^lS,  V5,  and  Vl. 

3.  V5  and  V6.  8.    V3,  V5,  and  ■\^2T- 

4.  ^7  and  VlO.  9.    Va&,  -^oF^  and  ■v/2. 

5.  VlO,  V2,  and  -y/B.  -  10.    Va,  ^6,  "\/a;,  and  V.y. 

6.  Vi,  V2,  and  V3.  11.    Va  +  6  and  Vic  +  2/- 

12.  Which  is  greater,  ^5  or  V2?   \/4  or  V3? 

13.  Which  is  greater,  a/3  or  a/4?  3V2  or  2a/4? 
Arrange  in  order  of  ascending  value : 

14.  V3,  V2,  and  ^7.  17.   V2,  a/S,  ^%  and  a/J. 

15.  V2,  a/4,  and  a/5.  18.    a/7,  ^48,  a/4,  and  ^63. 

16.  A/'2,  a/'3,  and  a/30.  19.    a/4,  a/5,  -^13,  ^150. 

ADDITION  AND  SUBTRACTION  OF  RADICALS 

311.  Radicals  that  in  their  simplest  form  are  of  the  same 
order  and  have  the  same  radicand  are  called  similar  radicals. 

Thus,  2  -n/S  and  7  a/3  are  similar  radicals. 

312.  Principle.  —  Only  similar  radicals  can  he  united  into 
one  term  by  addition  or  subtraction. 

EXERCISES 

313.  1.  Find  the  sum  of  V50,  2^8,  and  6a/i. 

Explanation.  —  To  ascertain  whether  the  given 

_  expressions  are  similar  radicals,  each  may  be  re- 

a/50  =    5a/2  duced  to  its  simplest  form.     Since,  in  their  simplest 

2-\/S  =    2 a/2  form,  they  are  of  the  same  order  and  have  the  same 

a^J  __    Q  ^^2  radicand,  they  are  similar,  and  their  sum  is  obtained 

■=:  by  prefixing  the  sura  of  the  coefficients  to  the  com- 

Sum  =  10 a/2  jjion  radical  factor. 


RADICALS 


223 


Find  the  sum  of : 

2.  V50,  Vl8,  and  V98.  5.    V28,  V63,  and  VTOO. 

3.  V27,  Vi2,  and  V75.  6.   ^'250,  ^l6,  and  ^4.. 

4.  V20,  V80,  and  V45.  7.    "v/m,  ^686,  and  \/|. 

8.  \/l35,  ^320,  and  \/625. 

9.  a/500,  \/i08,  and  V^^2. 

10.  VJ,  Vl2i,  Vi,  and  VH- 

11.  Vi  V75,  |V3,  and  V12. 

12.  V|,  JV3,  J</9,  and  Vli7. 

13.  ^40,  V28,  ^25,  and  Vl75. 

14.  Vl47,  4V20,  V75,  and  V605. 

15.  ^I92,  V80,  4V45,  and  5\/24. 
Simplify : 


16.  V245-V405H-V45. 

17.  V12  +  3V75-2V27. 

18.  5V72+3V18-V50. 

19.  ■v/128  +  \/686-\/54. 

20.  VlT2- V343  +  V448. 


„-        fa  ,      fa         fa 
22.    ,/^'_  Jll^. 


24.  V(a4-6)2c-  Via-hfc. 

25.  ■v/^-^^^6^+-^8^iW; 


26.    \/3i»3  +  30i»2  +  75aj- V3a^-6ic2-f  3a;. 


27.    V5a^  +  30a^  +  45a3- V5a''-40a^  +  80a«. 


28.  V50  +  V9-4ViH-V-24  4-V27-V64. 

29.  V|4-6V|-iVl8  +  ^36-^4f  +  ^i25-2VS. 


224  RADICALS 

MULTIPLICATION   OF   RADICALS 

314.  a^  X  a}  =  a^'^^'  =  a^'^^  ==  aK 
That  is,         Va  x  "v^a  =  Va^  x  Va^  =  'y/o^. 

Since  fractional  exponents  to  be  united  by  addition  must  be 
expressed  with  a  common  denominator,  radicals  to  be  united 
by  multiplication  must  be  expressed  with  a  common  root  index. 

EXERCISES 

315.  1.    Multiply  V7  by  V5;  5V3  by  2Vi5. 

PROCESSES 

V7x  V5  =  V7^=V35 
5  V3  X  2  Vis  =  5  X  2^3x15  =  10  V45  =  10V9  V5  =  30  V5 

2.  Multiply  2  V3  by  3^/2. 

PROCESS 

2  V3  =  2 .  3^  =  2  .  3^  =  2-s/27 
3^2  =  3-2^  =  3.2^  =  3^4 
2V3  X  3^2  =  2^27  x  3\/4  =  2  x  3^27x1:  =  6^108 

Rule.  —  If  the  radicals  are  not  of  the  same  order,  reduce  them 
to  the  same  order. 

Multiply  the  coefficients  for  the  coefficient  of  the  product  and  the 
radicands  for  the  radical  factor  of  the  product  ;  simplify  the  re- 
sult^ if  necessary. 

Multiply : 

3.  V2  by  Vs.  8.    2  V3  by  3^45. 

4.  V2  by  V6.  9.   2v/6  by  3V6. 

5.  V3  by  Vl5.  10.    3V3  by  2^5. 


6.  2 V5  by  3VI0.  11.   2V24  by  Vl8. 

7.  3  V20  by  2  V2.  12.    V2^  by  3  V^'. 


RADICALS  225 

Find  the  value  of :     - 
13.    Vmw  X  Vm^n.  x  Vmn*. 
«14.    V2  axy  X  ^/xy  x  ^a^xy. 

15.  V^^  X  -J/^  X  </(a-6)-l 

16.  V|  X  Vf  X  Vf .  .  18.   16^  X  2^  X  32l 

17.  ^  X  \/|  X  V|.  19.    27^  X  9i  X  8li 

20.  Multiply  2  V2  +  3  V3  by  5  V2  -  2  V3. 

Solution 

2\/2+3\/3 
6V2-2\/3 
20      +  15V6 

-    4\/6-18 
20      +ll\/«-18  =  2  + llVO. 
Multiply: 

21.  V5  +  V3  by  V5  —  V3. 

22.  V7  +  V2by  \/7- V2.     • 

23.  V6  -  V5  by  VO  -  VS. 

24.  5  —  V5  by  1  +  Vs. 

25.  4V7  +  lby  4V7  — 1. 

26.  2V2  +  V3by  4Vii+ V3. 

27.  a'  -  a6 V2  +  6^  ^y  ^2  _^  ^^y^  +  h\ 

28.  a;V^  — a;V2/4-2/V^  — 2/V</ by  Va;4- Vy. 
Expand : 

29.  (V3+V5)(V3-Vo)-    31.   (Vo  +  vri)(V6- VH). 

30.  (V9+V6)(V9-V6)-    32.  (V5a+aV5)(V5^'^vt). 

MILNE's    IST   YR.    ALG. — 15 


226  RADICALS 

DIVISION   OF  RADICALS 

316.  a^^a^  =  a^-^  =  a^^  =  aK 

That  is,     ^a^^/a=  </a'  h-  ^/'a'  =  </W^^=  -Va. 

In  division,  when  one  fractional  exponent  is  subtracted  from 
another,  the  exponents  must  be  expressed  with  a  common 
denominator.  When  one  radical  is  divided  by  another,  the 
radicals  must  be  expressed  with  a  common  root  index. 

EXERCISES 

317.  1.   Divide  V60  by  Vi2. 

PROCESS 

V60-j-Vl2  =  V60--12  =  V5 
2.   Divide  \/2  by  V2. 

PROCESS 

^-V2  =  ^^-^=  ^1=^  =  4^/32 


6¥ 

E.ULE.  —  If  necessary,  reduca  the  radicals  to  the  same  order. 

To  the  quotient  of  the  coefficients  annex  the  quotient  of  the  radi- 
cands  written  under  the  common  radical  sign,  and  reduce  the 
result  to  its  simplest  form. 

Find  quotients : 

3.  V5d^V8.  9.  ^l&^-s/M. 

4.  V72--2V6.  10.  V2aP--^/a¥. 

5.  4V5^V40.  11.  -C/aV--V2o^. 


6.  6V7-f-Vl26.  12.    V9a262^V3a6. 

7.  ^4-^V2.  13.    -y/^^^^^^xy. 


8.   7V75-f-5V28.    •  14.    Va  -  6 -^  Va  +  6. 


RADICALS  227 

15.  Divide  Vl5  -  V3  by  V3. 

16.  Divide  V6  -  2  V3  +  4  by  V2. 

17.  Divide  V2  +  2  +  4  V42  by  -|  V6. 

18.  Divide  5  V2  +  5  V3  by  VlO  +  VlS. 

19.  Divide5  +  5V30  +  36by  V5  +  2V6. 

INVOLUTION  AND   EVOLUTION   OF  RADICALS 

318.  In  finding  powers  and  roots  of  radicals,  it  is  frequently 
convenient  to  use  fractional  exponents. 

EXERCISES 

319.  1.   Find  the  cube  of  2  Vaac*. 

Solution.       (2  Va^y  =  28(0^ )«  =  8  a^x^  =  8  V^d^  =  8  ax'  Vox. 

2.   Find  the  square  of  3  Var^. 

Solution.         (3 y/s^)^  =  9  (Jy^  =  Qx^'  =  9  y/s^^  =  9x  Vx^. 


Square : 

Cube: 

Involve  as  indicated : 

3.   SVab. 

7.   2  Vs. 

11.    (-2V2a6). 

4.   2  ^3  a;. 

8.   3V2. 

12.    (-^/2^/xy. 

5.   xs/2  3^. 

9.    2-V^^ 

13.   (-V2V^^ 

6.    rv'VIb. 

10.    Va'6^ 

14.    (-2V^^yy. 

15.  Obtain  by  the  binomial  formula  the  cube  of  V2  4- 1. 

Solution.     CV2+  1)^=  (V2)8 +3  (\/2)2  •  1  +  3  V2  •  P  +  p 

=  2  \/2  +  6  +  3  a/2  +  1 

=  7  +  5  \/2. 
Expand : 

16.  (2  +  V6)2.  18.    (2+V5y.  20.    (V7-V6)2 

17.  (2  +  V2)2.  19.    (2-V3)^  21.    (2V2-V3)2. 


228  RADICALS 

22.  What  is  the  fourth  root  of  V2x ? 
Solution.  V  V2a  =  [(2x)^]i  =  (2x)^  =  i/2x. 
Find  the  square  root  of :  Find  the  cube  root  of : 

23.  V2.  25.    -v^^.  27.   V2^.  29.   -  27V^. 

24.  -v/S.  26.    ^P.  28.   </Sm^\     30.    -64\/^. 
Simplify  the  following  indicated  roots : 

31.   V^'4aV.  32.   ^|Vc^*.  33.    (VSaV)^ 

320.  A  binomial,  one  or  both  of  whose  terms  are  surds,  is 
called  a  binomial  surd. 

\/2  +  V5,  2  +  V5,  \/2  +  1,  and  VS  -  v^  are  binomial  surds. 

321.  A  binomial  surd  whose  surd  or  surds  are  of  the  second 
order  is  called  a  binomial  quadratic  surd. 

V2  +  Vs  and  2  +  V5  are  binomial  quadratic  surds. 

322.  Two  binomial  quadratic  surds  that  differ  only  in  the 
sign  of  one  of  the  terms  are  called  conjugate  surds. 

3  +  V5  and  3  —  V6  are  conjugate  surds  ;  also  VS  -\-  y/2  and  VS  —  V2. 

323.  The  square  root  of  a  binomial  quadratic  surd  by  inspection. 
The  square  of  a  binomial  may  be  written  in  the  form 

(a-\-by=:(a^-{-b^-h2ab. 
Thus,  (V2  +  V6)2  =  (2  +  6)+  2  VT2  =  8  +  2Vl2. 

Therefore,  the  terms  of  the  square  root  of  8  +  2  Vl2  may  be 
obtained  by  separating  Vl2  into  two  fa.ctors  such  that  the 
sum  of  their  squares  is  8.     They  are  V2  and  V6. 

That  is,  AI8  4-2VI2  ^■V2  +  V6. 

Principle.  —  The  terms  of  the  square  root  of  a  binomial 
quadratic  surd  that  is  a  perfect  square  may  be  obtained  by  divid- 
ing the  irrational  term  by  2  and  then  separating  the  quotient  into 
two  factors,  the  sum  of  whose  squares  is  the  rational  term. 


RADICALS  229 


EXERCISES 

324.    1.    Find  the  square  root  of  14  -f  8  V3. 
Solution 

14  +  8\/3  =  14+2(4V8)=  14  +  2\/48. 
Since  V48  =  \/6  x  VS  and  14  =  6  +  8, 


Vl4  +  8V3  =  Vd  -f  V8  =  V6  +  2 V2. 

2.    Find  the  square  root  of  11  —  6^/2. 

Solution 

Vll-6V2=Vll-2Vl8=  V9-  \/2=3- 

.V2. 

Find  the  square  root  of : 

3.    12  +  2\/35.               7.    11  +  2V30.            11. 

12  4-  4  Vs. 

4.    16-2V60.               8.    7-2V10.              12. 

11  +  4  V7. 

5.    15-h2V26.               9.    12-6V3.              13. 

15-6V6. 

6.    16-2V55.              10.    17  +  12V2.            14. 

18  +  6  V5. 

RATIONALIZATION 

325.   Suppose  that   it  is  required  to  find  the 

approximate 

due  of ,  having  given  V3  =  1.732  •••. 

V3 
1.732  ...|1.000000i.577  ...           3)1.732 
8660                                 .577  . 

— 

We  may  obtain  a  decimal  approximately  equal  to  — r,  as  in 

V3 
the  first  process  (incomplete),  by  dividing  1  by  1.732  •..;  but 
a  great  saving  of  labor  may  be  effected  by  first  changing  the 
fraction  to  an  equal  fraction  having  a  rational  denominator, 
thus : 

J_^   1.  V3   ^V3 
V3      V3  .  V3       3  ' 
and  employing  the  second  process. 


230  RADICALS 

326.  The  process  of  multiplying  an  expression  containing  a 
surd  by  any  number  that  will  make  the  product  rational  is 
called  rationalization. 

327.  The  factor  by  which  a  surd  expression  is  multiplied  to 
render  the  product  rational  is  called  the  rationalizing  factor. 

328.  The  process  of  reducing  a  fraction  having  an  irrational 
denominator  to  an  equal  fraction  having  a  rational  denomi- 
nator is  called  rationalizing  the  denominator. 


BXERCISBS 

329.   Find  the  value  of  each  of  the 

following  to  the  nearest 

third  decimal  place, 

taking  V2  =  1.41421, 

V3  =1.73205,  and 

V5  =  2.23607 : 

^•^• 

4.      3. 

V6 

7.      10    . 

V45 

^4 

^•^-        , 

8.      1^. 
V50 

3.     2. 

V5 

Vl2 

9.        1     . 
V125 

Eationalize  the  denominator  of  each  of  the  following,  using 
the  smallest,  or  lowest,  rationalizing  factor  possible : 


10.    ^.  14.    ?^.  18.    ^ 


Vb  Vby  '  Vr  +  1 


11.  J-.  15.    .^.  19.    ^^ 

V5^  ^12  -Va-b 

ax  Va 

12.  -:===•  16.    -r^-^-  20. 


■\ax^ 


\x-2 
^x  +  2' 


13.    ^^r^.  17.    yffyl.  21.    ^^"^^ 


TP  ^3aJ2/2  \(a-6) 


3    (^-^ 


RADICALS  231 

330.  The  product  of  any  two  conjugate  surds  is  rational. 

For,  by  §94,  (Va  +  V^)(Va-  V6)  =  (Va)2-  {y/b)^  =  a-b. 

Hence,  a  binomial  quadratic  surd  may  be  rationalized  by  mul- 
tiplying it  by  its  conjugate. 

EXERCISES 

9 

331.  1.   Rationalize  the  denominator  of 


3-V5 
Solution 

2       ^        2(3  +  V5)  ^  2(3  4-  V5)  ^  3  4  V5 

3_V6      (3-V5)(3+V5)  ^-5  2 

/» 
2.   Rationalize  the  denominator  of ^• 

V7+V3 

Solution 

6         _  6(V7-\/3)  ^6(v'7-  \/3)^3(\/7- \/3) 


V7  +  V3      (V7+  V3)(V7- V3)            7-3 
Rationalize  the  denominator  of : 
3.    -A_.  6.    — i 


2  +  V3  V3-V2                        a  +  2V6 

4.        ^  7.         ^                         10.    V|+V5 

V5-V3  *    2-V2                        "    Vx-Vi/ 

6.    — ? 8.    -^-5 11.   ^-^g. 

V3  +  V5  V6-2V3                     3-hVa6 

12.    1? 14.    — ^ 


3V3-2V2                            aj+VaJ^-l 
13.  ^-^       .  15      ^+J^ 


Va;  +  lH-2  Va;  +  y+Va;  — i/ 


Reduce  to  a  decimal,  to  the  nearest  hundredth  : 

16.    ^_.  17.    -^.  18.  g 

2-V3  3-1-V5  V3-V2 


232  RADICALS 


RADICAL  EQUATIONS 

332.   An  equation   involving   an  irrational  root   of  an  un- 
knovrn  number  is  called  an  irrational,  or  radical,  equation. 


x^  =  3,   Vx  +  1  =  Vx  —  4  +  1,  and   y/x 


1=2  are  radical  equations. 


333.  A  radical  equation  may  be  freed  of  radicals,  wholly  or 
in  part,  by  raising  both  members,  suitably  prepared,  to  the  same 
power.  If  the  given  equation  contains  more  than  one  radical, 
involution  may  have  to  be  repeated. 

When  the  following  equations  have  been  freed  of  radicals, 
the  resulting  equations  will  be  found  to  be  simple  equations. 
Other  varieties  of  radical  equations  are  treated  subsequently. 

EXERCISES 

334.  1.   Given  V2lc  +  4  =  10,  to  find  the  value  of  x. 

Solution 

V2x  +  4  =  10. 
Transposing,  V'2  x  =  6. 

Squaring,  2  x  =  36. 

.-.  x  =  18. 
Verification.  —  Substituting  18    for  x  in  the  given  equation   and 
(§  293)  considering  only  the  positive  value  of  \/2  x,  we  have  VSG  +  4  =  10 ; 
that  is,  10  =  10  ;  hence,  the  equation  is  satisfied  for  x  =  18. 


2.    Given  ^x  —  7  +  V.v  =  7,  to  find  the  value  of  x. 
Solution 


Vx  -  7  +  vx  =  7. 
Transposing,  Vx  —  7  =  7  —  Vx. 

Squaring,  x  -  7  =  49  —  14Vx  +  x. 

Transposing  and  combining,       14  Vx  =  56. 
Dividing  by  14,  Vx  =  4. 

Squaring,  x  =  16. 

Verification.     V16  -  7  +  Vl6=  V0+  vT6  =  3+4=7;  that  is,  7 »7. 


RADICALS  233 

3.    Solve,  if  possible,  the  equation  Va;  —  7  —  Va  =  7. 

Solution 

Transposing,  squaring,  simplifying,  etc., 
>/x=-4. 

Squaring,  a;  =  16. 

Verification.  —  Substituting  16  for  x  in  the  given  equation  and 
(§  293)  considering  only  the  positive  value  of  Vx  —  7  and  of  Vx,  the 
first  member  becomes 


Vl6  -7  -  Vl6  =  V9  -  Vl6  =  3-  4  =  -  1 ; 

but  the  second  member  of  the  given  equation  is  7  ;  hence,  x  =  16  does 
not  satisfy  the  equation. 

That  is,  the  equation  has  no  root,  or  is  impossible. 

General  Directions.  —  Transpose  so  that  the  radical  term,  if 
there  is  but  one,  or  the  most  comjylex  radical  term,  if  there  is  more 
than  one,  may  constitute  one  member  of  the  equation. 

Then  raise  each  member  to  a  power  corresponding  to  the  order 
of  that  radical  and  simplify. 

If  the  equation  is  not  freed  of  radicals  by  the  first  involution, 
proceed  again  as  at  first. 

Solve,  and  verify  results,  denoting  impossible  equations: 

4.    \/^Tl  =  3.  13.    Va;  +  16-Vx  =  2. 


5.  Va;  -h  5  =  4.  14.    V2^  -  V2  a;  -  3  =  1. 

6.  Vx~^  =  l.  15.    V2a-+-V2a;-  3  =  1. 


7.  Va;  -  a-  =  b.  16.    -^Jx^  -{-x  +  l  =  2  —  x. 

8.  v/«^^  =  2.  17.    3V^^^  =  3.r-3. 


9.  -Vx  -a^=:a.  18.  V 3  .'c  +  7  +  V3  a.-  =  7. 

10.  Vi  H-  6  =  a.  19.-  2  V«  +  V4a;  —  11  =  1. 

11.  l+\/a;  =  6.  20.  5~V.'M^=V«. 

12.  2Va  =  6-v^.  21.  Vx^  -5d;  +  7  +  2  =  a;. 


234  RADICALS 

Solve,  and  verify  results,  denoting  impossible  equations : 
22.   4:-V4:-Sx-]-9x^  =  Sx. 


23.    V3a;  — 5-h  V3a;  +  7  =  6. 


24.    \/4a;-|-5  — 2Vx— 1  =  9. 


25.    V2a;-1+ V2aj  +  4=5. 


26.    V5a;- 1-1  =  V5a;-f-16. 


27.    V^-h3V5a;-16-4  =  0. 


28.  2a; -^4  0^-  Vl6a^-7  =  1. 

29.  V^x  -  V^  =  V9a;-32. 

30.  ■\/2(x'  +  Sx-5)  =  (a;  +  2) V2. 


31.    V2(a7  +  1)4-V2a;-1=  V8a;-f  1. 


32.  Solve  the  equation  V2  x  —  V2  x  —  7  = 
Suggestion.  —  Clear  the  eq 

33.  Solve  the  equation 


V2a;-7 
Suggestion.  —  Clear  the  equation  of  fractions. 

V3^  +  15      V3^-f-6 


V3a;+    5      V3x  +  1 

Suggestion.  —  Some  labor  may  be  saved  by  reducing  each  fraction  to 
a  mixed  number  and  simplifying  before  clearing  of  fractions. 

Thus,  1  +      ^Q       =  1  +        ^ 


V3a:  +  5  V3a:  +  1 

Canceling,  and  dividing  both  members  by  5, 
2         ^        1 

\/3x  +  5      VSx  +  1 
Solve  and  verify : 


,^      Vs-1      Vs-3  '       ^^  V2¥+9      V2x  +  20 

Vs  +  5      Vs-1'  V2a;-7      V2a;-12 

35.    Vr:^  ^  V^"^^  37  V^  +  18^      32       ^  -^ 

Vm       V^^^  '  V»H-2        Va;>6 


REVIEW  236 


REVIEW 


335.    1.   Distinguish  between  involution  and  evolution. 

2.  Give  the  law  of  exponents  for  involution ;  for  evolution. 

3.  For  what  values  of  n  between  1  and  12  is  (—  2)"  posi- 
tive ?  negative  ?  What  is  the  sign  for  any  power  of  a  positive 
number  ? 

4.  How  is  a  fraction  raised  to  a  power  ?  How  is  the  root 
of  a  fraction  found  ?     Raise  -f^  to  the  second  power.     V^=  ? 

5.  How  is  the  power  of  a  product  found?  the  root  of  a 
product  ? 

6.  What  operation  is  indicated  by  a  radical  sign?  In 
what  other  way  may  this  operation  be  indicated  ?     Illustrate. 

7.  How  many  values  has  V25  ?  What  is  the  principal 
square  root  of  25  ?     What  is  the  principal  cube  root  of  —  8? 

8.  What  is  the  index  of  a  root?  What  index  is  meant 
when  none  is  expressed  ? 

9.  Distinguish  between  real  and  imaginary  numbers  and 
illustrate  each. 

10.  What  is  the  sign  of  an  odd  root  of  a  number  ?  of  an  even 
root  of  a  number  ? 

11.  What  is  the  meaning  oi  aP? 

12.  When  a  number  has  a  fractional  exponent,  what  does  the 
numerator  of  the  exponent  show  ?  the  denominator  ? 

13.  What  is  the  meaning  of  x-^  ?  How  may  any  factor  be 
transferred  from  one  term  of  a  fraction  to  the  other  ? 

Illustrate  by  writing  without  a  denominator :  — -  • 

14.  Expand  (x  —  yf  by  the  binomial  formula.  How  does 
the  number  of  terms  correspond  with  the  exponent  of  the 
power  ?  What  is  the  coefficient  of  the  first  and  last  terms  ? 
of  the  second  term?  How  are  the  coefficients  of  the  other 
terms  obtained  ? 


236  REVIEW 

15.  Define  and  illustrate  radical,  radicand,  entire  surd,  and 
mixed  surd. 

16.  What  is  meant  by  the  order  of  a  radical?     Illustrate 
by  giving  radicals  of  different  orders. 

17.  How  may  a  radical  of  the  second  order  be  represented 
graphically  ? 

Illustrate  by  representing  graphically  V26. 

18.  What  is  a  rational  number?  an  irrational  number? 
From  the  following  select  the  rational  numbers : 

8;  I;  V3;  </S',   -Tf;  ^25-,  5^ 


19.  Are  -y/TB  and  Vl  -f  V2  radicals  ?  surds  ?  Are  all 
radicals  surds  ?     Are  all  surds  radicals  ? 

20.  How  may  the  coefficient  of  a  radical  be  placed  under 
the  radical  sign  ? 

Express  as  entire  surds:  ^V2;   9^bc;  ^'\/^. 

21.  When  is  a  radical  in  its  simplest  form  ?• 

Illustrate  by  reducing  V40  b%  V|,  and  V4,  each  to  its  sim- 
plest form. 

22.  What  are  similar  radicals?  When  numbers  have  frac- 
tional exponents  with  different  denominators,  what  must  be 
done  to  the  fractional  exponents  before  the  numbers  can  be 
multiplied  ?     Find  the  value  of  5i  x  lOi  X  6i. 

23.  What  is  a  binomial  surd  ?  a  binomial  quadratic  surd  ? 
What  are  conjugate  surds  ? 

24.  Define  rationalization ;  rationalizing  factor. 

How  may  a  binomial  quadratic  surd  be  rationalized  ? 

7x 
Rationalize  the  denominator  of  —:=r 


Va4- V^ 

25.   What  is  a  radical  equation  ?     Give  general  directions 
for  solving  a  radical  equation. 

Solve  and  verify :  Va;  -h  V3-f  a;  =  3. 


QUADRATIC   EQUATIONS 


336.  The  equation  a;  —  2  =  0  is  of  the  first  degree  and  has  one 
root,  X  =  2.  Similarly,  ic  —  3  =  0  is  of  the  first  degree  and  has 
one  root,  x  =  3.  Consequently,  the  product  of  these  two  simple 
equations,  which  is 

(x-2){x-3)=0,ovx'-5x-^(y  =  0, 
is  of  the  second  degree  and  has  two  roots,  2  and  3. 

337.  An  equation  that,  when  simplified,  contains  the  square 
of  the  unknown  number,  but  no  higher  power,  is  called  an  equar 
tion  of  the  second  degree^  or  a  quadratic  equation. 

It  is  evident,  therefore,  that  quadratic  equations  may  be  of 
two  kinds  —  those  which  contain  only  the  second  power  of  the 
unknown  number,  and  those  which  contain  both  the  second  and 
first  powers. 

a;2  =  15  and  3  jc-^  +  2  ic  =  4  are  quadratic  equations. 

PURE  QUADRATIC  EQUATIONS 

338.  An  equation  that  contains  only  the  second  power  of  the 
unknown  number  is  called  a  pure  quadratic. 

2  x2  =  8  and  4  x^  —  2  a;^  =  16  are  pure  quadratic  equations. 

339.  The  equation  «^=  16  has  two  roots,  for  it  may  be  re- 
duced to  the  form  (x  —  4)  (x  +  4)  =  0,  which  is  equivalent  to 
the  two  simple  equations, 

X  —  4=0  and  a^  -|-  4  =  0, 
each  of  which  has  one  root,  namely,  +  4  and  —  4.     That  is, 

Principle.  —  Every  pure  quadratic  equation  has  two  roots, 
numerically  equal  but  opposite  in  sign. 

237 


238  QUADRATIC   EQUATIONS 

EXERCISES 

340.    1.   Given  10  x'  =  99  -  a^,  to  find  the  value  of  x. 

Solution 
10  a;2  =  99  -  a;2. 
Transposing,  etc.,  Ilaj2  =  99. 

Dividing  by  11,  ofi  =  9. 

Taking  the  square  root  of  each  member,  §  276, 

x=  ±S. 

Note.  —  Strictly  speaking,  the  last  equation  should  be  ±  x  =  ±  3, 
which  stands  for  the  equations,  +cc  =  +3,  +ic=  —  3,  and  —  x  =  —  3, 
and  —  X  =  +S.  But  since  the  last  two  equations  may  be  derived  from 
the  first  two,  by  changing  signs,  the  first  two  ^icpnss  all  the  values  of  x. 

For  convenience,  then,  the  two  expressions,  3C  =:  +  3  and  ic  =  —  3,  are 
written  x  =  ±  S. 

Consequently,  in  finding  the  square  roots  of  the  members  of  an  equa- 
tion, it  will  be  sufficient  to  write  the  double  sign  before  the  root  of  one 
member. 

2.  Find  the  roots  of  the  equation  3  a^  =  24. 

Solution 
3  x2  =  24. 
Dividing  by  3,  x^  =  8. 

Taking  the  square  root,  x=  ±  2\/2. 

Vebification.  —  The  given  equation  becomes  24  =  24  and  is  therefore 
satisfied  when  either  +  2  V2  or  —  2  \/2  is  substituted  for  x. 

Solve,  and  verify  each  result : 

3.  3a^-5  =  22.  10.  4:n^ -{-9  =  5n^ -7. 

4.  2aj2  +  3a;2  =  80.  11.  (a;  +  2)2- 4(a;  +  2)  =4. 

5.  4a^  =  f    .  12.  (Su-S)(Su-\-S)  =  17. 

6.  |x2-5  =  22.  13.  (x-^iy-{x-iy  =  3S. 

7.  5ar^-75  =  2£c^.  l^.'  (x  +  iy=.x(3x-\-2) -3. 

8.  2a;2-25  =  73.  15.  5(s +  2)  =:3s^  + s(5  - s). 

9.  70^^  =  43^  +  24.  16.  (2r-hl)(2r+3)=8(r-f3). 


QUADRATIC   EQUATIONS  239 

Problems 

341.  1.  The  length  of  a  10-acre  field  is  4  times  its  width. 
What  are  its  dimensions  ? 

2.  How  many  rods  of  fence  will  inclose  a  square  garden 
whose  area  is  2^  acres  ? 

3.  A  4-foot  length  of  stone  curbing  contains  3840  cubic 
inches.  If  its  width  is  5  times  its  thickness,  find  these 
dimensions. 

4.  The  width  of  the  largest  American  flag  is  |  of  its  length. 
The  area  of  the  flag  is  1500  square  feet.  What  are  the  dimen- 
sions of  the  flag  ? 

5.  A  farmer  sold  his  pumpkins  for  ^50.  The  number  of 
tons  was  8  times  the  number  of  dollars  he  received  per  ton. 
Find  the  number  of  tons  sold  and  the  price  per  ton. 

6.  A  lace  worker  in  Switzerland  received  $  12  for  a  piece  of 
work.  The  number  of  days  he  worked  on  it  was  -^  of  the 
number  of  cents  he  earned  per  day.  What  were  his  daily 
earnings  ? 

7.  The  area  of  the  face  of  a  square  drawing  board  is  5  square 
feet.  Eind  the  dimensions  of  a  rectangular  board  3  inches 
longer  and  3  inches  narrower,  if  its  face  has  the  same  area. 

8.  A  shipment  of  railroad  ties  measuring  400,000  board  feet 
contained  as  many  car  loads  as  there  were  board  feet  in  a  tie. 
If  each  car  held  250  ties,  find  the  total  number  of  ties  and  the 
number  of  board  feet  in  one  tie. 

Formulae 

342.  Solve  the  following  formulae  from  physics : 

1.  s  =  lgf,iovt.  ^    i^=^',foru 

2.  E  =  ^Mv',iovv.  ^ 


240 


QUADRATIC   EQUATIONS 


6.  When  g  =  32.16,  formula  1  gives  the  number  of  feet  (s) 
through  which  a  body  will  fall  in  t  seconds,  starting  from  rest. 
How  long  will  it  take  a  brick  to  fall  to  the  sidewalk  from  the 
top  of  a  building  100.5  feet  high  ? 

7.  To  lighten  a  balloon  at  the  height  of  2500  feet,  a  bag  of 
sand  was  let  fall.  Find  the  time,  to  the  nearest  tenth  of  a 
second,  required  for  it  to  reach  the  earth. 

Solve  the  following  geometrical  formulae : 

8.  c^  =  a^  +  h\  for  h.  10.   A  =  .7854  d\  for  d. 

9.  4  m2  =  2(a2  +  6')  -  c^,  f or  m.  11 .    F=  ^  Trrh,  for  r. 

12.  Using  formula  8,  find  the  hypotenuse  (c)  of  a  right  tri- 
angle whose  other  two  sides  are  a  =  8  and  &  =  6. 

13.  By  means  of  formula  8,  find  the  side  (a)  of  a  right 
triangle  whose  hypotenuse  (c)  is  5  and  whose  side  (6)  is  3. 

14.  From  formula  8  and  the  accompanying 
figure  find,  to  the  nearest  tenth,  the  side  (a) 
of  a  square  inscribed  in  a  circle  whose  diame- 
ter (d)' is  10. 

15.  Using  formula  8  and  the  accompany- 
ing figure,  deduce  a  formula  for  the  alti- 
tude Qi)  of  an  equilateral  triangle  in  terms 
of  its  side  (c). 

16.  From  formula  9,  find  the  length  of  the 
median  (m)  to  the  side  (c)  of  the  triangle  in 
the  accompanying  figure,  if  a  =  11,  6  =  8,  and 
c  =  9. 

17.  Substituting  in  formula  10,  find,  to 
the  nearest  tenth  of  a  foot,  the  diameter  {d) 
of  a  circle  whose  area  {A)  is  1000  square  feet. 

18.  Using  formula  11,  find,  to  the  nearest  centimeter,  the 
radius  (r)  of  the  base  of  a  conical  vessel  20  centimeters  high 
Qi  =  20)  that  will  hold  a  liter  of  water  (F=  1  liter  =  1000  cu. 
cm.;  7r  =  3.1416). 


QUADRATIC   EQUATIONS  241 

AFFECTED  QUADRATIC  EQUATIONS 

343.  A  quadratic  equation  that  contains  both  the  second  and 
the  first  powers  of  one  unknown  number  is  called  an  affected 
quadratic. 

yi  +  Sx  =  \0  and  4x^  —  x=S  are  affected  quadratics. 

344.  To  solve  affected  quadratics  by  factoring. 

Reduce  the  equation  to  the  form  ax^ -\- bx -\- c  =  0,  factor  the 
first  member,  and  equate  each  factor  to  zero,  as  in  §  163,  thus 
obtaining  two  simple  equations  together  equivalent  to  the  given 
quadratic. 

Thus,  3a:2  =  10a;-3. 

Transposing,  3a;2_i0ic-|-3=0. 

Factoring,  (a;  —  3)  (3  a;  —  1)  =  0. 

.-.  a:  — 3  =  0or3jc-l  =  0; 
whence,  r  =  3  or  \. 

EXERCISES 

345.  Solve  by  factoring,  and  verify  results : 

1.  a^-5x  +  e  =  0.  13.  20^-7  x-\-S  =  0. 

2.  a^ 4- 10^  +  21  =  0.  14.  2z^--z-S  =  0. 

3.  .^2-1-12  a;- 28  =  0.  15.  3  ir'- 2  v-S  =  0. 

4.  x^-20x-\-51=0.  16.  10r2-27r4-5  =  0. 

5.  af-5x=:2i.  17.  6(524.!)  =  13s. 

6.  a^-l  =  3(ar-|-l).  18.  2x^-\-7x  =  4:. 

7.  .'c2  +  10a;  =  39.  .    19.  4  0.-2  +  6=110;. 

8.  eO-^x'^llx.  20.  Sw'-\-Sw  =  6. 

9.  a;(a;-l)  =  42.  21.  2^(^  +  3)+4  =  0. 

10.  a^-8  =  2(x-\-6).  22.    S^x" -2)-7  x  =  0. 

11.  a^-ll(.T-h3)  =  9.  23    42/24-82/4-3  =  0. 

12.  5iy  +  x(x  +  16)  =  0.  24     10(2  -  3  a;  +  a:^)  ^  q^ 

MILNE's    IST    YR.    ALG.  —  10 


242  QUADRATIC   EQUATIONS 

346.  First  method  of  completing  the  square. 

Since  .  (x  -\- af  =  x^ -{■  2  ax -j-  a% 

the  general  form  of  the  perfect  square  of  a  binomial  is 

x^-{-2ax-{-  a\ 

Consequently,  an  expression  like  x^  +  2a,x  may  he  made  a 
perfect  square  by  adding  the  term  a^,  which  it  will  be  observed 
is  the  square  of  half  the  coefficient  of  x. 

Thus,  to  solve  a^  -j-  6  a?  =  —  5 

by  the  method  of  taking  the  square  root  of  both  members  (the 
method  used  in  solving  pure  quadratics),  we  must  complete  the 
square  in  the  first  member. 

The  number  to  be  added  is  the  square  of  half  the  coefficient  of 
x;  that  is,  (f  )^,  or  9.  The  sayne  number  must  he  added  to  the 
second  member  to  preserve  the  equality. 

Therefore,  Ax.  1,  x'^ -\- 6x +  9  =- 5  +  9'y 

that  is,  a^  H- 6a; -f  9  =  4. 

Taking  the  square  root,  §  275,  a;  +  3  =  ±  2; 
whence,  a;  =  —  3  -f-  2  or  —  3  —  2. 

.*.  a;  =  —  1  or  —  5. 

EXERCISES 

347.  1.     Solve  the  equation  a;^  —  5  a;  — 14  =  0. 

Solution 
a;2  -  5  a;  -  14  =  0. 
Transposing,  oj^  —  5  x  =  14. 

Completing  the  square,         x^  —  5  oj  +  -^^  =  14  +  ^  =  8^. 
Taking  the  square  root,  a;  —  |  =  i  | ; 

whence,  a;  =  |  +  |or|—  f. 

.'.  X  =  7  or  —  2. 

Verification.  —  Either  7  or  —  2  substituted  for  x  in  the  given  equation 
reduces  it  to  0  =  0  ;  that  is,  the  given  equation  is  satisfied  by  these  values 
of  X. 


QUADRATIC   EQUATIONS  243 

2.   Solve  the  equation  4a^-f4a;  —  2  =  0. 

Solution 
4a;2  +  4x-2  =  0. 
Transposing,  4  a;^  +  4  ic  =  2. 

Dividing  by  4,  x^  +  x  =  ^. 

Completing  the  square,  x^  +  x  +  \  =  ^-\-\=^. 

Taking  the  square  root,  x-\-  1  =  ±^  V3. 

.-.  X  =  -  I  +  ^  V3  or  -  ^  -  1  Vs; 
which  would  usually  be  written,  x  =  ^(—  1  ±  VS). 

Steps  in  the  solution  of  an  affected  quadratic  equation  by  the 
first  method  of  completing  the  square  are : 

1.  Transpose  so  that  the  terms  containing  o^  and  x  are  in  one 
member  and  the  known  terms  in  the  other. 

2.  Make  the  coefficient  of  x^  positive  unity  by  dividing  both 
members  by  the  coefficient  ofx^. 

3.  Complete  the  square  by  adding  to  each  member  the  sqvAire  of 
half  the  coefficient  of  x. 

4.  Find  the  square  root  of  both  members. 

5.  Solve  the  two  simple  equations  thus  obtained. 

Solve,  and  verify  all  results : 

3.  a^-4a;  =  6.  12.  /  =  10-32/. 

4.  a^-6a;  =  7.  13.  ir^ -ir  ^  v  =  14:, 

5.  8a;  =  ar^-9.  14.  n(n-V)  =  2. 

6.  a;2  +  2aj  =  15.  15.  -^2  4-3^  =  1. 

7.  x'  -2x  =  24.  16.  2a; (a;  -  2)  =  8. 

8.  a;2  +  8a.-  =  -15.  17.  r2  +  4r-7=0. 

9.  63  =  «"^  +  2a;.  18.  Z^  _  n  ;  _^  28  =  0. 

10.  ic2-12a;=-ll.  19.    Sar' -  3a; -2  =0. 

11.  a;2_4o=:6a;.  20.   3a;--6a?=-2. 


244  QUADRATIC   EQUATION JS 

348.   Hindoo  method  of  completing  the  square. 

EXERCISES 

1.  Solve  the  general  quadratic  equation  ax^  -^bx-\-c  =  0. 

Solution 

ax^  +  bx  +  c  =  0.  (1) 

Transposing  c,  .  ax^  +  bx  =  —  c.  (2) 

Multiplying  by  a,  a^x^  +  abx  =  -  ac.  (.3) 

Completing  the  square,       a%2  ^  (^,5^  +  ^  =  ^  _  ^c^  r^\ 

4       4 
Multiplying  by  4,  4  d^x'^  +  4  a6:«  +  &2  _  52  _  4  ^^^  ^^^ 

Taking  the  square  root,  2ax-\-b=  ±  Vb^  -  4'ac.  (6) 

2« 

It  is  evident  that  (5)  can  be  obtained  by  multiplying  (2)  by  4«  and 
adding  b^  to  both  members.  Hence,  when  a  quadratic  has  the  general 
form  of  (1),  if  the  absolute  term  is  transposed  to  the  second  member,  as 
in  (2),  the  square  may  be  completed  and  fractions  avoided  by 

Multiplying  by  4  times  the  coefficient  of  y?-  and  adding  to  each  member 
the  square  of  the  coefficient  of  x  in  the  given  eq%iation. 

This  is  called  the  Hindoo  method  of  completing  the  square. 

Note.  —  The  formula  for  the  values  of  a;,  given  in  (7)  and  known  as 
the  quadratic  formula,  may  be  used  in  obtaining  the  roots  of  any  quad- 
ratic by  substituting  the  numerical  values  of  «,  &,  and  c  found  in  the  given 
equation  after  it  is  reduced  to  the  form  ax'^  -\- bx -\- c  =  0. 

Solve  by  the  Hindoo  method,  then  by  the  quadratic  formula : 

2.  3ic2+4a;  =  4.  9.  5a;2-7a;=-2.               ^ 

3.  2aj2_lla;4.l2  =  0.  .  10.  ^x^-\-^x=-l. 

4.  5a^-14a;=-8.  11.  2 -\-^x+2x' =  0. 

5.  2aj2  +  5aj  =  7.  12.  &x^-{-2  =  lx. 

6.  2a^  +  7a;=-6.  13.  4.xr  +  ^x  =  ^. 

7.  3a^-7a;=-2.  14.  8a?  +  2ar^  =  9. 

8.  4ar»-a;-3  =  0.  15.  15  i^- 7.^- 2  =  0. 


QUADRATIC   EQUATIONS  245 

349.   Miscellaheous  equations  to  be  solved  by  any  method- 

EXERCISES 

General  Directions.  —  1.    Reduce   the  equation   to   the  general 
form  a^ -\-hx-\-c=^0. 

2.  If  the  factoids  are  readily  seen,  solve  by  factoring. 

3.  If  the  factors  are  not  readily  seen,  solve  by  completing  the 
square  or  by  formula. 

Note.  — In  reducing  fractional  equations  to  the  general  form,  observe 
the  cautions  given  on  page  146. 

Solve,  and  verify  each  result : 

1.   0^4.5  =  60;.  ,_      X    ,  x^  —  15     X 

15-   Tr.+  -ir        - 


2.  ^2  =  3  a;  4- 10. 

3.  ox-\-3x^  =  2.  16. 

4.  2a^-7a;  =  2. 


12         5  a;         5 

_  X X  —  o  _  3 

.v-o         X     ~2' 

x±±,^_(x±S)^ 


5.  a:2_i2a;  =  28.  17.   ^__  +  3=^_^ 

6.  x*-16=zx'(x'-l).  ^  4 

7.  a^- 13  a; -30  =  0.  ^^*   a;-2'"a;-2'^^' 

8.  .x-^  +  4'(a;  -  3)  =  0.  19        a;         l^a;  +  2 

a*  4-  2       ^^         2  rr    * 

9.  6  +  lla;  +  3a^  =  0.  ^         ^        ^"^ 

10.  (a; +  5)2  =  10  a; +  74.  20.    -^  +  eiil|  =  3. 
^          ^  .r  +  7      a;  +  3 

11.  4  a^- 3  a; -2  =  0. 

;r  +  2_a;  +  5_. 

12.  (a;  +  7)(a;-9)=l-2a;.      ^^'    x-7     ^^  ~ 

13.  a?  +  l-^  =  0.  22.    ^/~^  =  2-        "" 


a;     2  a:2_3^  x2-3a; 

14.  ^        ^x-2  23     a;-2      x  +  2_     40 


9(a;-l)         6  a;  +  2     2-a;     a;--4 

Find  roots  to  two  decimal  places: 

24.   ar'_4a;-l=0.  26.    ^2^5^  +  5.6  =  0. 

26.   v«+6'y  +  7  =  0.  27.   t' - 12  t +  16.5  =  0. 


246  QUADRATIC   EQUATIONS 

Literal  Equations 

350.  The  methods  of  solution  for  literal  quadratic  equations 
are  the  same  as  for  numerical  quadratics.  The  method  by 
factoring  (§  344)  is  recommended  when  the  factors  can  be 
seen  readily.  If  it  is  necessary  to  complete  the  square,  the 
first  method  (§  346)  is  usually  more  advantageous,  provided  the 
coefficient  of  a^  is  + 1,  otherwise  the  Hindoo  method  (§  348) 
is  better,  because  by  its  use  fractions  are  avoided.  Results 
may  be  tested  by  substituting  simple  numerical  values  for  the 
literal  known  numbers. 

EXERCISES 

351.  Solve  for  x  by  the  method  best  adapted : 

1.  x^  —  ax  =  ab  —  bx.  4.   5  a?  —  2  aa;  =  a^  — 10  a. 

2.  x^-\-ax  =  ac-\-cx.  5.   a^  +  3  6a;  =  5  ca;  +  15  be. 

3.  xr=(m  —  n)x  +  mn.  6.    6x^ -{-3  ax  =  2bx-\-ab. 

7.  acx^  —  bcx  —  bd-{-  adx  =  0. 

8.  a;^-f  4  ma;  4-3  na;  4-12  mw  =  0. 

9.   x^  =  4.ax-2a\  ^^    x  +  ~  =  --^b. 

o     ^  X      b 

10.  ar  —  aa;  —  a^  =  0. 

3/p2 

11.  Aax-x'  =  Sa\  18.    2a; =  a-2a;. 

a 

12.  5ax-^ea^=6x^.  .  . 

19.  _J_  =  1-^^^. 

13.  21b'-4:bx  =  x'.  ax  +  4.  16 

20.  ^  +  ^a;  =  ^. 
b  b 

21.  a;^4-2  =  [^^'"^^la?. 


14. 

7m2 
12 

x" 
-ma;  =  -. 

15. 

x' 
36 

5  a; 
4 

-1 

1  A 

X 

X 

^) 


m. 


22.   ^_2x^4(a6-l). 


a;  _  1     a;  4- 1  a6  ab 


QUADRATIC   EQUATIONS  247 

23.  y?  —  2{a  —  h)x=^ab. 

24.  x^  —  2  x(m  —  w)  =  2  mn. 

25.  a:^-\'2(a  +  S)x=-S2a. 

26.  x'-i-x-^-bx-^-b^aix  +  l). 

27.  a(2x-l)+2bx-b  =  x(2x-l). 

28.  a^  +  4(a-l)x=8a-4al 

29.  _-^  =  l  +  Ul. 

a-f  o  +  ic      a      0     ic 

2a-a;      rt  +  2a;~3' 
31.   a(x  —  2a-\-b)-\-a(x-^a  —  b)  =  x^—(a  —  bf. 

RADICAL   EQUATIONS 

352.  In  §§  333,  334,  the  student  learned  how  to  free  radical 
equations  of  radicals,  the  cases  treated  there  being  such  as  lead 
to  simple  equations.  The  radical  equations  in  this  chapter 
lead  to  quadratic  equations,  but  the  methods  of  freeing  them 
of  radicals  are  the  same  as'in  the  cases  already  discussed. 

EXERCISES 

353.  1.    Solve  the  equation  2  ViP  —  x  =  x  —  S  -Vx. 

Solution 
2Vx  —  x  =  x—S  Vx. 
Dividing  by  Vx,  2  —  Vx  =  Vx  —  8. 

Transposing,  etc.,  Vx  =  6. 

Squaring,  x  =  26. 

Verification.  —  When  x  =  25, 

1st  member  =  2  V26  —  25  =  10  -  25  =  -  16  ; 
2d  member  =26-8  V25  =  25  -  40  =  -  15. 
Hence,  x  =  25  is  a  root  of  the  equation  ;  x  =  0,  the  root  of  the  equa- 
tion Vx  =  0,  also  is  a  root  of  the  given  equation,  removed  by  dividing 
both  members  by  Vx. 


248  QUADRATIC   EQUATIONS 

0. 


2.    Solve  and  verify  -y/x  +  1  -f  Va;  - 

-2- 

-V2a;- 

5 

Vx  +  1  +  Vx 

Solution 

0. 

-  2  -  V2  X  - 

-5  = 

Transposing, 

\/ 

a:  +  1  +  Vx  - 

^  = 

V2X-5. 

Squaring,        a 

c  +  1  +2\/ic2- 

-X-2+X- 

■2  = 

2x-5. 
-2. 

Simplifying, 

Vx2  -  X  - 

-2  = 

Squaring, 

X2-X- 

•2  = 

4. 

Solving, 

x  = 

-  2  or  3. 

Verification. 

,  —  Substitutin 

g  —  2  for  X  in  the 

given  equf 

itic 

V- 1  +  V34  -  V- 9  =  0 ; 
that  is,  V-^  +  2V^-3V^^  =0. 

Therefore,  —  2  is  a  root  of  the  given  equation. 
Substituting  3  for  x  in  the  given  equation, 

Vi  4-  Vl  -  Vl  =  0, 

which  is  not  true  according  to  the  convention  adopted  in  §  293. 

Hence,  3  is  not  to  be  regarded  as  a  root  of  the  given  equation. 

Note.  — The  equation  could  be  verified  for  x  =  3,  if  the  negative  square 
root  of  1  were  taken  in  the  second  terra  and  the  positive  square  root  in 
the  third,  thus : 

Vi  +  Vl  -  Vl  =  2  +  (-  1)  -  (  +  1)  =  0. 

This  is  an  improper  method  of  verification,  however,  for  it  has  been 
agreed  previously  that  the  square  root  sign  shall  denote  only  the  positive 
square  root. 

Solve,  and  verify  each  result : 

3.  8  V^  —  8  a;  =  f .  5.   a;  —  1  +  Va;  +  5  =  0. 

4.  3  a;  -f-  Vx  =  5  vTaJ.  6.   x—  5  —  Va;  —  3  =  0. 

7.    V4  a;  +  17  -f  V^T~i  -4  =  0.' 


8.    l+V(3-5a;)2-f  16  =  2(3-a;). 


9.    VI  -f-  x^o^  +  12  =  1  -f-  a;. 
10.    Va?  —  1  H-  V2  a;  —  1  -  V5^ 


QUADRATIC   EQUATIONS  249 

11.  V2a;-  7  -  V2^+Va;  -  7  =0. 

12.  Vci  +  ic  —  Va  —  X  =  V2~«. 


13.    Va;  —  a  -f-  Vft  —  a;  =  V6  —  a. 


14.   V2a;-fvT0^TT  =  V2a;H-l. 


15.    V6  4-if+Va;-Vl0-4a;  =  0. 


16.    V4a;-3-V2ic-f-2=Vx-6. 


17.  V2  «  + 3— Va;+1  =V5a;  — 14. 

18.  VxMTs -^ =  x. 

Var^  +  8 


19.    Va  —  a:  +  V6  —  a;  =  Va  +  6  —  2  a;. 


Find  roots  to  two  decimal  places : 


20.  x  —  \/(yx=6.  22.   2  +  2 Va?  +  3  +  .t  =  0. 

21.  3  V^  =  2(a;  -  2).  •    23.    Vx'^^  +  V2'a;  =  3. 

Problems 

354.     1.   The  sum  of  two  numbers  is  8,  and  their  product  is 
15.     Find  the  numbers. 

Solution.  —  Let  x  =  one  number. 

Then,  8  —  x  =  the  other. 

Since  their  product  is  16,  (8  —  x)x  =  15. 
Solving,  sc  =  3  or  5, 

and  8  —  a;  =  6  or  3. 

.  Therefore,  the  numbers  are  3  and  6. 

2.  Divide  20  into  two  parts  whose  product  is  96. 

3.  Divide  14  into  two  parts  whose  product  is  45. 

4.  Find  two  consecutive  positive  integers  the  sum  of  whose 
squares  is  61. 


250  QUADRATIC   EQUATIONS 

Solve  the  following  problems  and  verify  each  solution : 

5.  A  plumber  received  $  24  for  some  work.  The  number  of 
hours  that  he  worked  was  20  less  than  the  number  of  cents  per 
hour  that  he  earned.     Find  his  hourly  wage. 

6.  A  man's  life  insurance  for  a  certain  time  cost  $20.80. 
If  the  number  of  weeks  was  12  more  than  the  number  of  cents 
he  paid  per  week,  what  was  the  weekly  premium  ? 

7.  The  area  of  a  revolving  floor  in  a  hall  in  Paris  is  2650 
square  feet.  The  width  is  3  feet  less  than  the  length.  What 
are  the  dimensions  of  the  floor  ? 

8.  The  base  of  the  tower  of  the  Metropolitan  Life  Building 
in  New  York  City  is  6375  square  feet  in  area.  The  length  of 
the  base  is  10  feet  greater  than  the  width.  What  are  the  di- 
mensions of  the  base  ? 

9.  The  1860  bunches  of  asparagus  from  an  acre  of  land 
were  sold  in  boxes  each  holding  1  less  than  ^  as  many  bunches 
as  there  were  boxes.     Find  the  number  of  bunches  in  a  box. 

10.  One  of  the  largest  electric  signs  in  the  world  is  14,560 
square  feet  in  area.  The  length  of  the  sign  lacks  22  feet  of 
being  twice  the  width.     Find  the  dimensions  of  the  sign. 

11.  The  area  of  the  plate  glass  floor  of  the  highest  bridge  in 
the  world,  built  across  Royal  Gorge  in  Colorado,  is  5060  square 
feet.  The  length  is  10  feet  more  than  10  times  its  width. 
What  is  its  length  ? 

12.  If  the  average  amount  deposited  in  the  postal  savings 
banks  of  Canada  by  each  depositor  one  year  had  been  $70 
less,  and  if  the  number  of  depositors  had  been  equal  to  the  aver- 
age number  of  dollars  each  deposited,  the  total  deposit  would 
have  been  $  40,000.     Find  the  average  amount  each  deposited. 

13.  The  length  of  a  steel  barge  used  for  coal  on  the  Ohio 
Eiver  is  5  feet  more  than  5  times  its  width.  If  the  depth  is 
8  feet  and  the  capacity  of  the  barge  is  28,080  cubic  feet,  what 
is  the  width  ?  the  length  ? 


QUADRATIC   EQUATIONS  251 

14.  The  area  of  the  plate  used  in  a  giant  camera  is  37| 
square  feet.  The  width  of  the  plate  is  8  inches  more  than  ^  the 
length.     Find  the  dimensions  of  the  plate. 

15.  The  Chester,  a  United  States  scouting  cruiser,  steamed 
106.12  knots  on  its  trial  voyage.  The  number  of  knots  that 
it  went  in  one  hour  was  1.47  less  than  7  times  the  number  of 
hours  spent  on  its  trial  voyage.     Find  its  speed  per  hour. 

16.  A  piece  of  silk  made  from  spiders'  web  and  exhibited 
at  the  Paris  Exposition  was  36  times  as  long  as  it  was  wide. 
If  its  width  had  been  increased  9  inches,  it  would  have  con- 
tained V6^  square  yards.     Find  its  length  and  width. 

17.  The  area  of  one  side  wall  of  a  square  reservoir,  cut  in  the 
solid  rock  at  Bowling  Green,  Ohio,  is  2200  square  feet,  and 
the  depth  of  the  reservoir  is  2  feet  more  than  \  of  its  length. 
Find  its  three  dimensions. 

18.  Some  boys  laid  out  basket-ball  grounds  30  feet  greater 
in  length  than  in  width,  but  to  change  the  area  to  the  pre- 
scribed limit  of  3500  square  feet,  they  reduced  the  length 
10  feet.     How  much  too  large  had  they  laid  out  the  grounds  ? 

19.  A  party  hired  a  coach  for  $12.  In  consequence  of  the 
failure  of  3  of  them  to  pay,  each  of  the  others  had  to  pay  20 
cents  more.     How  many  persons  were  in  the  party  ? 

Solution 
Let  X  =  the  number  of  persons. 

Then,  a;  —  3  =  the  number  that  paid, 

12 
—  =  the  number  of  dollars  each  should  have  paid, 

X 

12 
and =  the  number  of  dollars  each  paid. 

Therefore,       -^ -=1^. 

x-3     5      a; 

Solving,  X  =  15  or  —  12. 

The  second  value  of  x  is  evidently  inadmissible,  since  there  could  not 
be  a  negative  number  of  persons. 

Hence,  the  number  of  persons  in  the  party  was  16. 


252  QUADRATIC   EQUATIONS 

20.  A  club  had  a  dinner  that  cost  $  60.  If  there  had  been 
5  persons  more,  the  share  of  each  would  have  been  $  1  less. 
How  many  persons  were  there  in  the  club  ? 

21.  A  party  of  young  people  agreed  to  pay  $  8  for  a  sleigh 
ride.  As  4  were  obliged  to  be  absent,  the  cost  for  each  of  the 
rest  was  10  cents  greater.     How  many  went  on  the  ride  ? 

22.  Find  two  consecutive  integers  the  sum  of  whose  recip- 
rocals is  ^. 

23.  The  dry  gum  used  per  day  in  gumming  United  States 
postage  stamps  costs  $  48.  If  400  pounds  more  were  used  at 
the  same  total  cost,  the  price  per  pound  would  be  2  cents  less. 
How  much  gum  is  used  daily  ? 

24.  A  man  earned  $  48  by  shearing  sheep.  The  number  of 
cents  he  earned  per  fleece  was  2  more  than  the  number  of  days 
he  worked.  How  many  sheep  did  he  shear,  if  he  averaged  100 
a  day  ? 

25.  The  weight  of  80  four-inch  spikes  was  3  pounds  liBss 
than  the  weight  of  80  five-inch  spikes.  If  1  pound  of  the 
former  contained  6  spikes  more  than  1  pound  of  the  latter, 
how  many  of  each  kind  weighed  1  pound? 

26.  A  tub  of  dairy  butter  weighed  20  pounds  less  than  a  tub 
of  creamery  butter,  and  360  pounds  of  dairy  butter  required 
3  tubs  more  than  the  same  amount  of  creamery  butter.  What 
weight  of  butter  was  there  in  a  tub  of  each  kind  ? 

27.  Mr.  Field  paid  $  8  for  one  mile  of  No.  9  steel  wire 
and  f  2.88  for  one  mile  of  No.  14,  wire.  The  No.  9  wire 
weighed  224  pounds  more,  and  cost  ^  cent  per  pound  less, 
than  the  No.  14  wire.     Find  the  cost  of  each  per  pound. 

28.  A  boy  earned  $  420  by  delivering  bills.  If  he  had  re- 
ceived 50  cents  more  per  thousand,  he  would  have  earned  as 
much  by  delivering  70  thousand  less  than  he  did.  How  much 
was  he  paid  per  thousand  ? 


QUADRATIC   EQUATIONS  25B 

29.  A  merchant  sold  a  hunting  coat  for  $  11,  and  gained  a 
per  cent  equal  to  the  number  of  dollars  the  coat  cost  him. 
What  was  his  per  cent  of  gain  ? 

30.  A  moving  picture  film  150  feet  long  is  made  up  of  a 
certain  number  of  individual  pictures.  If  these  pictures  were 
\  of  an  inch  longer,  there  would  be  600  less  for  the  same 
length  of  film.     How  long  is  each  separate  picture  ? 

31.  In  Detroit,  a  machine  for  making  pills  turned  out 
250,000  pills  per  hour.  If,  in  boxing  these,  5  pills  more  were 
put  into  each  box,  the  number  of  boxes  would  be  2500  less. 
How  many  pills  does  each  box  contain  ? 

32.  Two  coats  of  paint  applied  to  the  sides  of  a  barn  having 
an  area  of  195  square  yards  required  69  pounds  of  paint.  One 
pound  covered  1^  square  yards  more  for  the  second  coat  than 
for  the  first.  What  area  did  1  pound  of  paint  cover  for  each 
coat  ? 

33.  A  train  started  16  minutes  late,  but  finished  its  run  of 
120  miles  on  time  by  going  5  miles  per  hour  faster  than  usual. 
What  was  the  usual  rate  per  hour  ? 

34.  Two  automobiles  went  a  distance  of  60  miles,  one 
making  6  miles  per  hour  faster  time  than  the  other  and  com- 
pleting the  journey  ^  of  an  hour  sooner.  How  long  was  each 
on  the  way  ? 

35.  The  distance  covered  by  an  aeroplane  on  one  occasion 
was  45  miles.  If  its  rate  per  minute  had  been  ^  of  a  mile 
more,  the  distance  would  have  been  covered  in  9  minutes  less 
time.     Find  the  speed  of  the  aeroplane  per  minute. 

36.  If  each  cable  of  the  Manhattan  Bridge  contained  9 
strands  more  and  each  strand  20  wires  more,  the  number  of 
wires  in  a  strand  would  be  6  times  as  many  as  the  number  of 
strands  in  a  cable.  How  many  strands  are  there  in  a  cable,  if 
there  are  9472  wires  in  a  cable  ? 


254  QUADRATIC   EQUATIONS 

37.  To  run  around  a  track  1320  feet  in  circumference  took 
one  man  5  seconds  less  time  than  it  took  another  who  ran  2 
feet  per  second  slower.     How  long  did  it  take  each  man  ? 

38.  Some  rugs  made  in  India  have  400  knots  to  the  square 
inch.  A  fast  weaver  ties  40  knots  more  per  minute  than  a  boy. 
If  the  former  weaves  a  square  inch  in  13^  minutes  less  time 
than  the  latter,  how  many  knots  does  each  tie  per  minute  ? 

39.  A  cistern  can  be  filled  by  two  pipes  in  24  minutes.     If 

it  takes  the  smaller  pipe  20  minutes  longer  to  fill  the  cistern 

than  the  larger  pipe,  in  what  time  can  the  cistern  be  filled  by 

each  pipe  ? 

Solution 

Let  X  =  the  number  of  minutes  required  by  the  larger  pipe. 

Then,    a;  +  20  =  the  number  of  minutes  required  by  the  smaller  pipe. 

Since  -  =  the  part  that  the  larger  pipe  fills  in  one  minute, 

X 

=  the  part  that  the  smaller  pipe  fills  in  one  minute, 

and  -^  =  the  part  that  both  pipes  fill  in  one  minute. 

Then,  1  +  _L^=:_L. 

ic     x+20     24 

Solving,  a;  =  40  or  —  12. 

The  negative  value  is  inadmissible.  Hence,  the  larger  pipe  can  fill 
the  cistern  in  40  minutes,  and  the  smaller  pipe  in  60  minutes. 

40.  A  city  reservoir  can  be  filled  by  two  of  its  pumps  in 
3  days.  The  larger  pump  alone  would  take  1-|  days  less  time 
than  the  smaller.     In  what  time  can  each  fill  the  reservoir  ? 

41.  A  company  owned  two  plants  that  together  made  25,200 
concrete  building  blocks  in  12  days.  Working  alone,  one  plant 
would  have  required  7  days  more  time  than  the  other.  What 
was  the  daily  capacity  of  each  plant  ? 

42.  The  number  of  strawberry  baskets  made  by  a  machine 
was  12  more  per  minute  than  the  number  of  peach  baskets 
made  by  another  machine.     One  day  the  former  machine  started 
45  minutes  after  the  latter,  but  each  finished  2400  basketajafg 
the  same  instant.     Find  the  rate  of  each  per  minute.  :  .^^^^^ 


QUADRATIC   EQUATIONS 


265 


Formulae 

355.    1.   In  any  right-angled  triangle  (Fig.  1),  c^  =  a^-{-h^. 
Find  all  sides  when  a  =  c  —  2  and  6  =  c  —  4. 


Fig.  3. 

2.  The  area  {A)  of  a  triangle  (Fig.  2)  is  expressed  by  the 
formula  A  —  ^  ah.  If  the  altitude  (/i)  of  a  triangle  is  2  inches 
greater  than  the  base  (a)  and  the  area  is  60  square  inches, 
what  is  the  length  of  the  base  ? 

3.  If  two  chords  intersect  in  a  circle,  as  shown  in  Fig.  3, 
a  X  6  is  always  equal  to  cxd.  Compute  a  and  h  when  c  =  4, 
(Z  =  6,  and  6  =  a  4-  5. 

4.  The  formula  ^  =  a  +  v^  —  16  ^^  gives,  approximately,  the 
height  Qi)  of  a  body  at  the  end  of  t  seconds,  if  it  is  thrown 
vertically  upward,  starting  with  a  velocity  of  v  feet  per  second 
from  a  position  a  feet  high. 

Solve  for  t,  and  find  how  long  it  will  take  a  skyrocket  to 
reach  a  height  of  796  feet,  if  it  starts  from  a  platform  12  feet 
high  with  an  initial  velocity  of  224  feet  per  second. 

5.  How  long  will  it  take  a  bullet  to  reach  a  height  of 
25,600  feet,  if  it  is  fired  vertically  upward  from  the  level  of 
the  ground  with  an  initial  velocity  of  1280  feet  per  second  ? 

6.  When  a  body  is  thrown  vertically  downward,  an  approxi- 
mate formula  for  its  height  is  h  =  a  —  vt  —  l^  f,  in  which  h,  a, 
V,  and  t  stand  for  the  same  elements  as  in  exercise  4. 

Solve  for  t,  and  find  when  a  ball  thrown  vertically  downward 
from  the  Eiffel  tower,  height  984  feet,  with  an  initial  velocity 
of  24  feet  per  second,  will  be  868  feet  above  ground. 

Find,  to  the  nearest  second,  when  it  will  reach  the  ground. 


GRAPHIC  SOLUTIONS 


QUADRATIC  EQUATIONS  —  ONE  UNKNOWN  NUMBER 

356.    Let  it  be  required  to  solve  graphically,  cc^  —  6a;  +  5=0. 

To  solve  the  equation  graphically,  we  must  first  draw  the 
graph  of  a^ - 6 a;  +  5.     To  do  this,  let  y  =  a^-6x-\-5. 

The  graph  of  y  =  a:^  —  6x-{-5  will  represent  all  the  corre- 
sponding real  values  of  x  and  of  ay^  —  6x-\-5f  and  among  them 
will  be  the  values  of  x  that  make  a^  —  6x-\-5  equal  to  zero, 
that  is,  the  roots  of  the  equation  x^—6x-\-o  =  0. 

When  the  coefficient  of  aj^  is  -f- 1,  as  in  this  instance,  it  is 
convenient  to  take  for  the  first  value  of  ic  a  number  equal  to 
half  the  coefficient  of  x  with  its  sign  changed.  Next,  values 
of  X  differing  from  this  value  by  equal  amounts  may  be  taken. 

Thus,  first  substituting  x  =  S,  it  is  found  that  ?/  =  —  4,  locating  the 
point  J.  =  (3,  —4).  Next  give  values  to  x  differing  from  3  by  equal 
amounts,  as  2|  and  3-^,  2  and  4,  1  and  5,  0  and  6.  It  will  be  found  that 
y  has  the  same  value  for  x  =  Sl  as  for  x  =  2^,  for  a;  =  4  as  for  x  =  2,  etc. 

The  table  below  gives  a  record  of  the 
points  and  their  coordinates  : 


1 

1 

' 

2 

;E 

- 

i 

i 

i'l 

><l 

/ 1 

0 

D^ 

lMP  1 

<i ) 

b'Q 

\ 

r\ 

c? 

^C 

,^ 

' 

[ 

i 

bH 

_J 

_ 

y 

=  a;2-6a;  + 

5 

X 

y 

Points 

3 

-4 

A 

^,^ 

-3f 

B,  B' 

2,  4 

-3 

C,  C 

1,5 

0 

D,D' 

0,6 

5 

E,E' 

Plotting  the  points  A\  B,  B' \  C,  C ;  etc.,  whose  coordinates  are 
given  in  the  preceding  table,  and  drawing  a  smooth  curve  through  them, 
we  obtain  the  graph  of  y  =  a;^  -  6  a;  +  5  as  shown  in  the  figure. 

256 


GRAPHIC   SOLUTIONS 


267 


It  will  be  observed  from  the  work  of  the  preceding  page 
that: 

When  x  =  3jX^—6x-\-5  =  —4:,  which  is  represented  by  the 
negative  ordinate  PA. 

When  x  =  2  and  also  when  a;  =  4,  a^  —  6a;H-5=—  3,  which 
is  represented  by  the  equal  negative  ordinates  MG  and  NC. 

When  x  =  0  and  also  when  x  =  6,  a^  — 6a;  +  5  =  5,  repre- 
sented by  the  equal  positive  ordinates  OE  and  QE'. 

Thus,  it  is  seen  that  the  ordinates  change  sign  as  the  curve 
crosses  the  a>axis. 

At  D  and  at  D',  where  the  ordinates  are  equal  to  0,  the  value 
oi  ac^—6x-\-5  is  0,  and  the  abscissas  are  x—1  and  x=5. 

Hence,  the  roots  of  the  given  equation  are  1  and  5. 

The  curve  obtained  by  plotting  the  graph  of  ar^  —  6  ic  +  5,  or 
of  any  quadratic  expression  of  the  form  ajr^  +  6jr+c,  is  a 
parabola. 

EXERCISES 

357.     1.    Solve  graphically  the  equation  o^  —  8  a;  + 14  =  0. 

Solution.  —  Since  the  coefficient  of  x  is  —  8,  §356,  first  substitute  4 
for  X.     Points  and  their  coordinates  are  given  in  the  table : 


y 

=  a;2  -  8  a:  -f  14. 

X 

y 

Points 

4 

-2 

A 

3,  5 

-1 

B,  B' 

2,  6 

2 

c,  a 

1,7 

7 

D,D' 

A 


,  Plotting  these  points  and  drawing  a  smooth  curve  through  them,  we 
have  the  graph  of  y  =  x^  —  8  x  +  14,  which  crosses  the  x-axis  approxi- 
mately at  x  =  2.6  and  x  =  5.4. 

Hence,  to  the  nearest  tenth,  the  roots  of  x^  —  8  x  -f  14  =  0  are  2.6 
and  6.4. 

MILNE's    IST   YR.    ALG.  —  17 


268 


GRAPHIC   SOLUTIONS 


2.    Solve  graphically  the  equation  a^  —  8  ic  -f  16  =  0. 

Solution.  —  Since  the  coefficient  of  x  is  —  8,  §356,  first  substitute  4 
for  X.    Points  and  their  coordinates  are  given  in  the  table : 


y 

=  a;2_8x4-16 

X 

y 

Points 

4 

0 

A 

3,5 

1 

B,  B> 

2,6 

4 

C,  C 

1,7 

9 

D,  D< 

1 

' 

' 

Dl 

i° 

/ 

/ 

I 

C 

\ 

if 

\ 

A* 

B 

^ 

/ 

B 

0 

A 

1 

Plotting  these  points  and  drawing  a  smooth  curve  through  them,  we 
have  the  graph  oi  y  =  x^  —  Sx  +  16,  which  touches  the  ic-axis  at  x  =  4. 

This  fact  is  interpreted  graphically  to  mean  that  the  roots  of  the  equa- 
tion x^  —  Sx  +  16  =  0  are  equal,  both  being  represented  by  the  abscissa 
of  the  point  of  contact. 

Hence,  the  roots  are  4  and  4. 

Note.  —  The  student  may  show  that  4  and  4  are  the  roots  by  solving 
the  equation  algebraically. 

3.    Solve  graphically  the  equation  a.-^  —  8  ic  -f- 18  =  0. 

Solution.  —  Since  the  coefficient  of  a;  is  —  8,  §356,  first  substitute  4 
for  X.    Points  and  their  coordinates  are  given  in  the  table : 


i -dr — [fb- 

A 

0 


y 

=  a;2-8a;  +  18 

X 

y 

Points 

4 

2 

A 

3,5 

3 

B,  B' 

2,6 

6 

C,  C 

1,7 

11 

D,D'      , 

Plotting  these  points  and  drawing  a  smooth  curve  through  them,  we 


GRAPHIC   SOLUTIONS  259 

have  the  graph  of  y  =  x^  —  S  x  +  18^  which  neither  crosses  nor  touches 
the  X-axis.  This  fact  is  interpreted  graphically  to  mean  that  the  roots  of 
the  equation  x^  —  8  x  +  18  =  0  are  imaginary. 

Note.  —  The  student  may  show  that  the  roots  are  imaginary  by  solving 
the  equation  algebraically. 

In  eaxjh  of  the  preceding  graphs,  the  point  A,  whose  ordinate 
is  the  least  algebraically  that  any  point  in  the  graph  has,  is 
called  the  minimum  point. 

When  the  coefficient  of  ic^  is  -f  1,  it  is  evident  that : 

Principles.  —  1.  If  the  minimum  point  lies  below  the  x-axis, 
the  roots  are  real  and  unequal. 

2.  If  the  minimum  point  lies  on  the  x-axis,  the  roots  are  real 
and  equal. 

3.  If  the  minim^im  point  lies  above  the  x-a^is,  the  roots  are 
imaginary. 

Solve  graphically,  giving  real  roots  to  the  nearest  tenth : 

4.  x^-Ax-\-Z==0.  9.    x^-2x-2  =  0. 

'     5.  a?-(Sx  +  l  =0.  10.    x^  =  &x-^. 

6.  a^-4a;=-2.  11.   a.-^  +  4 a;  +  2  =  0. 

7.  a?  +  2{x-\-V)=0.  12.   ar'-2a;  +  6  =  0. 

8.  a^-4a;  +  6=0.  13.   a^-4a;-l  =  0. 

14.  Solve  graphically  4  a;  —  2  a.-^  + 1  =  0. 

Suggestion.  —  On  dividing  both  members  of  the  given  equation  by  —  2, 
the  coefficient  of  x^,  the  equation  becomes 

a:2  _  2  X  -  i  =  0. 

The  roots  may  be  found  by  plotting  the  graph  of  y  =  x^  —  2  x  —  ^. 

Solve  graphically,  giving  real  roots  to  the  nearest  tenth: 
15    2.^2  +  8.^  +  7  =  0.  17.    12a;-4a^-l=0. 

16.    2ar^-12a;  +  15  =  0.  18.    11 +8a;  +  2a,-2  =  0. 

Note.  —  Another  method  of  solving  quadratic  equations  graphically  is 
given  in  §  379. 


EQUATIONS  IN  QUADRATIC  FORM 


358.  An  equation  that  contains  but  two  powers  of  an  unknown 
number  or  expression,  the  exponent  of  one  power  being  twice 
that  of  the  other,  as  oa?^"  +  6a;"  -|-  c  =  0,  in  which  n  represents 
any  number,  is  in  the  quadratic  form. 

EXERCISES 

359.  1.   Given  a;^-10i»2  4- 24  =  0,  to  find  the  value  of  a;. 

Solution 
a4  _  10  a;2  +  24  =  0. 
Factoring,  {x'^  —  4)  (ic^  _  6)  =  0. 

Hence,  ic^  _  4  _  q  or  ic^  -  6  =  0  ; 

whence,  x=  ±2  ot:  x=  ±  V6. 

Each  of  these  values  substituted  in  the  given  equation  is  found  to  verify  ; 
hence,  ±  2  and  ±  V6  are  roots  of  the  equation. 

2.   Given  a;^—  oj^  —  6  =  0,  to  find  the  values  of  x. 

First  Solution 

cc^  -  x^  -  6  =  0. 

Let  X*  =  m,  then  x^  =  m^  and  the  equation  becomes 

w2  -  w  -  6  =  0. 

Solving  by  factoring,  m  =  3  or  —  2  ; 

that  is,  jc^  =  3  or  -  2. 

Raising  to  the  fourth  power,        a;  =  81  or  16. 

260 


EQUATIONS  IN  QUADRATIC   FORM  261 

Second  Solution 

Transposing,  x^  —  x^  =  6. 

Completing  the  square,  x^  —  x^  +  (i)2  =  -^. 

Taking  the  square  root,  x*  —  ^  =  ±  |. 

.-.  a;^  =  3or  -2. 
Raising  to  the  fourth  power,  x  =  81  or  16. 

Since  x  =  16  does  not  satisfy  the  given  equation,  16  is  not  a  root  and 
should  be  rejected. 

Solve,  and  verify  each  result : 

3.  x*-13a^  +  36  =  0.  9.  x^-x^  =  6, 

4.  a;^- 18^2 +  32  =  0.  lO.  a;  +  2Va;  =  3. 

5.  a;*-14a;2-f-45  =  0.  n.  a;^  _  a;^ - 12  =  0. 

6.  2a;^-10a^  +  8=0.  12.  (a;-3)2-|-2(a;- 3)=3. 

7.  3a;*-llar^  +  8  =  0.  13.  (x' - ly - ^x" - 1)  =  5. 

8.  a;i-5a;i  +  6=0.  14.  (a^-4)2-3(a^ -4)  =  10. 


15.   Solve  the  equation  o^  _  7  a;  +  Var'  -  7  a;  + 18  =  24. 
Solution 


x^- 
Adding  18  to  both  ra( 

-7x 
3mbe 

+  18 

:  + Vx2-7a;  +  18  =  24. 
rs,  we  have 

-7. 
-2. 

0) 

x^-Tx 

(  +  Va;'-^  -  7  X  +  18  =  42. 

and  m2  for  x2  -  7  x  +  18. 
m2  +  m  -  42  =  0. 

m  =  6  or 

(2) 

Put  m  for  Va;2  -  7  X  - 
Then,  transposing, 
Solving  by  factoring, 

f  18 

(3) 
(4) 

That  is. 

Vx2-7x  +  18  =  6, 

(5) 

or 
Squaring  (5), 
Solving, 

Vx2  -  7  X  +  18  =  -  7. 
x2-7x  +  18=36. 

X  =  9  or  - 

(6) 

Since,  in  accordance  with  §  293,  the  radical  in  (6)  cannot  equal  a  negative 
number,  Vx2  _  7  x  +  18  =  —  7  is  an  impossible  equation. 
Hence,  the  only  roots  of  (1)  are  9  and  —  2. 


262  EQUATIONS   IN   QUADRATIC   FORM 


16.    Solve  the  equation  x  +  2V«  +  3  =  21,  and  verify  results. 


Solution 

a^_9x3+8  =  0. 

Factoring, 

(a;3-l)(a;3-8)=0. 

Therefore, 

a:8-l=0, 

or 

a;3  -  8  =  0. 

17.  Solve  a;2  _  3  a;  4-  2 Va^  -  3  a;  +  6  =  18,  and  verify  results. 

18.  Solve  the  equation  x^  —  9  a^^  +  8  =  0. 


(1) 
(2) 
(3) 

(4) 

If  the  values  of  x  are  found  by  transposing  the  known  terms  in  (3)  and 
(4)  and  then  taking  the  cube  root  of  each  member,  only  one  value  of 
X  will  be  obtained  from  each  equation.  But  if  the  equations  are  factored, 
three  values  of  x  are  obtained  for  each. 

Factoring  (3) ,  (a;  -  1)  (a;2  +  a;  +  1)  =  0,  (5) 

and  (4),  (ic  -  2) (x2  +  2  a;  +  4)  =  0.  (6) 

Writing  each  factor  equal  to  zero,  and  solving,  we  have  : 

From  (5),  a^  =  1,  |(-  1  +  v^ITf),  l{-\-  yf^l).  (7) 

From  (6),  a;  =  2,  -  1  +  V^^,  -  1  -  V^.  (8) 

Note.  —  Since-  the  values  of  x  in  (7)  are  obtained  by  factoring 
a^  —  1  =  0,  they  may  be  regarded  as  the  three  cube  roots  of  the  number 
1.  Also,  the  values  of  x  in  (8)  may  be  regarded  as  the  three  cube  roots 
of  the  numbers  (^^271). 

Solve : 

19.   a;«-28a;3^27=0.  20.  a;^-16  =  0. 

21.  Find  the  three  cube  roots  of  —  1. 

22.  Find  the  three  cube  roots  of  —  8. 
23.   Solve  the  equation  a;*  +  4  a:^  —  8  a;  +  3  =  0. 

Solution 

By  applying  the  factor  theorem  (§146),  the  factors  of  the  first  mem- 
ber are  found  to  be  a;  —  1,  a;  +  3,  and  a;^  -f  2  x  —  1  ;  that  is, 

(«-l)(a;  +  3)(a;2  +  2a;-l)  =0. 
Solving,  a;  =  1,  -  3,  -  i  ±  V2. 


EQUATIONS  IN   QUADRATIC   FORM  263 

Solve,  and  verify  each  result : 

24.  x'-{-x'-4.x  =  -2.  26.    a^-8ic2  +  5a;+6  =  0. 

25.  x*-4:3^+Sx=-S.  27.   a;^-6a^  +  27a;  =  10. 

28.  a;*  +  6a^4-7a^-6a;-8  =  0. 

29.  a;*4-2ic3-10a-2-lla;  +  30  =  0. 

MISCELLANEOUS  EXERCISES 

360.    Solve,  aud  verify  each  result : 

1.  0.^-250:^4-144  =  0.  ^    x^-2o:^-3  =  0. 

2.  a;^-45or'  +  500  =  0.  e.   a'-3a;^=-2. 

3.  2o;*-lla^  +  12  =  0.  7.   a;^-2a;*  =  3. 

4.  5a;*-24ar'4-16  =  0.  8.   ic^-lOx* -f  9  =  0. 

9.    (iB2_l)2_4(ar^-l)+3  =  0. 

10.  (af^_6)2-7(a;2-6)-30  =  0. 

11.  (3(^-2xy-2{a:^-2x')  =  S. 

12.   a;-6a;^  +  8  =  0.  16.   a;-5  +  2Va;-5  =8. 


13.  a;4-20-9V^=0.  17.   2«- 3V2a;  +  5= -5. 

14.  2a;^-3a;^+l  =  0.  18-    2  aj-6V2o;-l  =  8. 

15.  ^t_7^i^io  =  0.  1^-.  ^-3  =  21-4V^33; 

20.   iB^-10a^  +  35ar'-50a;+24  =  0. 

21.       4:X'-4:X^-7X^  +  4:X  +  S  =  0. 

22.    16a;^-8ar^-31a^^-8.^'  +  15=0. 


23.  ic2-a;-Va;--a;-|-4-8  =  0. 

24.  a;2-5a;  +  2Va;--5a;-2  =  10. 


SIMULTANEOUS  QUADRATIC  EQUATIONS 


361.  Two  simultaneous  quadratic  equations  involving  two 
unknown  numbers  generally  lead  to  equations  of  the  fourth 
degree,  and  therefore  they  cannot  usually  be  solved  by  quad- 
ratic methods. 

However,  there  are  some  simultaneous  equations  involving 
quadratics  that  may  be  solved  by  quadratic  methods,  as  shown 
in  the  following  cases. 

362.  When  one  equation  is  simple  and  the  other  of  higher  degree. 

Equations  of  this  class  may  be  solved  by  finding  the  value 
of  one  unknown  number  in  terms  of  the  other  in  the  simple 
equation,  and  then  substituting  that  value  in  the  other  equation. 


363.    1.    Solve  the  equations  \        ^       ' 


Solution 

x  +  y  =  1. 

(1) 

-jS.-. 

3a;2  +  ?/2:^43. 

(2) 

From  (1), 

y  =  7  —  a. 

(3) 

Substituting  in  (2), 

3a;2.f-(7_«)2  =  43. 

(4) 

Solving, 

a;  =  3  or  ^. 

(6) 

Substituting  3  for  x  in 

(3),                     2/ =  4. 

(6) 

Substituting  \  for  x  in 

(3),                y  =  ¥- 

(7) 

That  is,  X  and  y  each  have  two  values 

'  when  a;  =  3,  y  =  4, 
when  iK  =  ^,  y  =  Y  • 

264 


SIxMULTANEOUS   QUADRATIC   EQUATIONS         265 

Solve,  and  verify  results : 

^      (x^  +  f  =  20,  ^      (x^  +  xy  =  12, 

\x=2y.  '     [x  —  y  =  2. 

(10x-^y  =  Sxyy  fm^-S^i^rzzlS, 


y-x  =  2. 


[m-2n  =  L 


4      f^  =  6-2/,  (3x(y-\-l)=12, 

[x'-hf  =  72.  '     [3x  =  2y. 


(xy(x-2y)=10,  ^      n 

[xy  =  10.  '    1: 


xy(x  —  2y)=10,  ^      (3rs  —  10r  =  s, 

2  -  s  =  -  r. 


364.  An  equation  that  is  not  affected  by  interchanging  the 
unknown  numbers  involved  is  called  a  symmetrical  equation. 

2x^  +  xy  +  2y^  =  4:  and  x^-\-y^  =  9  are  symmetrical  equations. 

365.  When  both  equations  are  symmetrical. 

Though,  equations  of  this  class  may  usually  be  solved  by 
substitution,  as  in  §§  362,  363,  it  is  preferable  to  find  values 
f or  a;  +  y  and  x  —  y  and  then  solve  these  simple  equations  for 
X  and  y. 


EXERCISES 

1.    Solve  the  equations 


xy  =  10. 


Solution 

x  +  y  =  7.  (1) 

xy  =  10,  (2) 

Squaring  ( 1 ) ,  x^ +2xy -\-y^  =  i9.  (3) 

Multiplying  (2)  by  4,  ixy=z  40.  (4) 

Subtracting  (4)  from  (S),  x^-2xy -hy^  =  9.  (5) 

Taking  the  square  root,  x  —  y  =  ±S.  (6) 

From  (1)  + (6),  a;  =  5  or  2. 

From(l)~(6),  y  =  2or5. 


266        SIMULTANEOUS   QUADRATIC   EQUATIONS 

r  a^  _l_  y2  __  25 
2.   Solve  the  equations  \  ' 


Squaring  (2), 
Subtracting  (1)  from  (3), 
Subtracting  (4)  from  (1), 
Taking  the  square  root, 
From  (2)  +  (6), 
From  (2)  -  (6), 


Solution 

^2  +  2/2  _  26. 

x^-\-2xy-\-  y^  =  49. 
2xy  =  24. 
ofi-2xy  ■\-y^=  1. 

x-y  =  ±\. 
a;  =  4  or  3. 


4. 


5. 


Solve,  and  verify  each  result : 
3     1^.^  +  2/^  =  50, 

x-\-y  =  ^, 
ic2  +  i/2  =  34. 

a;  -h  ?/  =  9, 
a;3  +  2/'  =  243. 

ic^  4- 0^  +  2/2  =  49. 

fa^  +  a.2/  +  2/'  =  31, 
iB=^  +  2/2  =  26. 


8. 


9.     ^ 


10. 


11. 


12. 


y  =  3  or  4. 


a^  +  /  =  8, 

I  ^2/ =  12, 

I  a;  —  a;?/  +  ?/  =  —  5. 

a^  +  3a;?/  +  ?/2  =  31, 
xy=e>. 

a^  +  /  =  100, 
196. 


ra^+/=i 

U^  +  2/)'  = 


a^  +  /  =  13, 
la^  +  2/2^a.'2/  =  19. 


(1) 
(2) 
(3) 
(4) 
(5) 
(6) 


367.  An  equation  all  of  whose  terms  are  of  the  same  degree 
with  respect  to  the  unknown  numbers  is  called  a  homogeneous 
equation. 

x^  —  xy  =  y^  and  Sa^  -\-y^  =  0  are  homogeneous  equations. 

An  equation  like  a^  —  xy-\-y^  =  21i8  said  to  be  homogeneous  in 
the  unknown  terms. 


SIMULTANEOUS  QUADRATIC  EQUATIONS         267 

368.  When  both  equations  are  quadratic  and  homogeneous  in  the 
unknown  terms. 

Substitute  vy  for  a;,  solve  for  y-  in  each  equation,  and  com- 
pare the  values  of  if  thus  found,  forming  a  quadratic  in  v. 

EXERCISES 

{x^-xy-^y^  =  21, 

369.  1.   Solve  the  equations  1   „     ^  . . 

[y^  —  2xy=  —16, 


Solution 

a;2-a;y  +  y2  =  21. 

(1) 

2/2  -  2  xy  =  -  15. 

(2) 

Assume 

x  =  vy. 

(3) 

Substituting  in  (1), 

«2y2_ry2  +  y2  =  21. 

(4) 

Substituting  in  (2), 

y2-2ry2=:_15. 

(5) 

Solving  (4)  for  y\ 

t/2-         21 

(6) 

^          tj2  -  t>  +  1 

Solving  (5)  for  y'^. 

«2-         15       . 

(7) 

^   -2t;-l 

Comparing  the  values  of  y"^ 

15                 21 

(8) 

2  r  -  1      «2  _  t>  +  1 

Clearing,  etc., 

6t 

|2-19t?  +  12=r0. 

(9) 

Factoring, 

(r- 

-  3)(5  r  -  4)  =  0. 

(10) 

.-.  r  =  3or^. 

(11) 

Substituting  3  for  v  in 

(7) 

or  in  (6),    y=  ±  V3    j 

(12) 

and  since  x  =  vy^  x  =  ±  3  v3.  J 

Substituting  f  for  t?  in  (7)  or  in  (6),     2/  =  ±  5  | 
and  since  x  =  vy,  x  =  ±  iJ 

When  the  double  sign  is  used,  as  in  (12)  and  in  (13),  it  is  understood 
that  the  roots  shall  be  associated  by  taking  the  upper  signs  together  and 
the  lower  signs  together. 


(x=SVB;         -3V3;        4;  -4 
[y=    V: 

Solve,  and  verify  results : 

.xy-f  =  S.  "    I  ic2  +  y2  =  125. 


Hence,  j  -  _ 

{y=    VS;         -  V3;  5 ;  -  5. 


ra^  +  3?/2  =  84,  r2a^-3a:2/  +  2/  =  100, 


268         SIMULTANEOUS   QUADRATIC   EQUATIONS 


^      ^a^-xy-^f  =  21, 


5. 


6. 


U2  +  22/'  =  27. 
'  x(x-y)=6, 
.ic2  +  2/2  =  5. 
xy^Sf  =  20y 
[af-Sxy  =  -S. 


^      (x'-^xy=.12, 
.xy  +  2y^  =  5. 

^      (x^-\-2f==U, 
[xy  —  y^  =  S. 

^      (a^-\-xy  =  77, 
I  xy  —  y^  =  12. 


370.  Special  devices. 

Many  systems  of  simultaneous  equations  that  belong  to  one 
or  more  of  the  preceding  classes,  and  many  that  belong  to  none 
of  them,  may  be  solved  readily  by  special  devices.  It  is  impos- 
sible to  lay  down  any  fixed  line  of  procedure,  but  the  object 
often  aimed  at  is  to  find  values  for  any  two  of  the  expressions, 
x-^y,  x  —  y,  and  xy,  from  which  the  values  of  x  and  y  msLj  be 
obtained.  Various  manipulations  are  resorted  to  in  attaining 
this  object,  according  to  the  forms  of  the  given  equations. 


371.    1.   Solve  the  equations 


EXERCISES 

x^-\-xy  =  12, 


Solution 

x2  +  iC?/  =  12. 

xy  -{-y^  =  4. 
Adding  (1)  and  (2),      x^-\-2xy  +  y^  =  16. 

.-.  X  -\-  y  = -\-  4:  or  — 4 
Subtracting  (2)  from  (1),         x^-y'^  =  S. 
Dividing  (5)  by  (4),  ic  -  y  =  +  2  or  -  2 

Combining  (4)  and  (6), 


(1) 

(2) 
(3) 
(4) 
(5) 
(6) 


a;  =  3or— 3;  2/  =  lor  —  1. 


Note.  — The  first  value  otx  —  y  corresponds  only  to  the  first  value  of 
x  +  y,  and  the  second  value  only  to  the  second  value. 

Consequently,  there  are  only  two  pairs  of  values  of  x  and  y. 

Observe  that  the  given  equations  belong  to  the  class  treated  in  §  368. 
The  special  device  adopted  here,  however,  gives  a  much  neater  and  sim- 
pler solution  than  the  method  presented  in  that  case. 


SIMULTANEOUS  QUADRATIC   EQUATIONS         269 

2.   Solve  the  equations    |  ^  +  2/'  +  aJ  +  ^  =  14, 

Solution 

X'2  +  y2  +  X  +  y  =  U.  (1) 

xy  =  3.  (2) 

Adding  twice  the  second  equation  to  the  first, 

x^-\-2xy  +  y2  +  x  +  y  =  20. 
Completing  the  square,  (x  +  yy  +  (x  +  y)  -\-  (^)2  =  20|. 
Taking  the  square  root,  x-\-y-\-^=±^. 

.'.  x  +  y  =  4oT  —  5.  (3) 

Equations  (2)  and  (3)  give  two  pairs  of  simultaneous  equations, 

\^-^y  =  ^  and  l^  +  y=-^ 

lxy  =  3  [xy  =  S 

Solving,  the  corresponding  values  of  x  and  y  are  found  to  be 

x  =  3;  1;  K-5+>/l3);  i(-5->/l3);       . 
y  =  l;  3;  i(_5-Vl3);  i(_6  +  VT3). 

Sjnnmetrical  except  as  to  sign.  —  When  one  of  the  equations 
is  symmetrical  and  the  other  would  be  symmetrical  if  one  or 
more  of  its  signs  were  changed,  or  when  both  equations  are  of 
the  latter  type,  the  system  may  be  solved  by  the  methods  used 
for  symmetrical  equations  (§  365). 


3.   Solve  the  equations    |^  +  2/'-53, 
[x  —  y  =  5. 

Solution 

x^  +  y^  =  53. 

(1) 

X  -  y  =  6. 

(2) 

Squaring  (2) ,                        x^-2xy-hy^  =  25. 

(3) 

Subtracting  (3)  from  (1),                    2xy  =  28. 

(4) 

Adding  (4)  and  (1),             x^ -{- 2xy -\- y^  =  S\. 

(6) 

Taking  the  square  root,                      x  +  y  =  ±9. 

(6) 

From  (6)  and  (2),                                      x  =  7  or  -  2  ; 

and                                                                 y  =  2  or  -  7. 

4.    Solve  the  equations 


270         SIMULTANEOUS   QUADRATIC   EQUATIONS 

Division  of  one  equation  by  the  other. — The  reduction  of  equa- 
tions of  higher  degree  to  quadratics  is  often  effected  by  divid- 
ing one  of  the  given  equations  by  the  other,  member  hy  member. 

x-y=2. 
Suggestion.  —  Dividing  the  first  equation  by  the  second, 

x'^  +  xy -{- y"^  =  13. 
Therefore,  solve  the  system, 

r  x2  +  a:?/  -H  2/2  =  13, 

U  -  y  =  2, 
instead  of  the  given  system. 

Elimination  of  similar  terms.  —  It  is  often  advantageous  to 
eliminate  similar  terms  by  addition  or  subtraction,  just  as  in 
simultaneous  simple  equations. 


5.    Solve  the  equations 


{xy  +  x  =  3^j 


Solution 

•      xy-\-x  = 

:35.                                              (1) 

xy  +  y  = 

32.            ^                            (2) 

Subtracting  (2)  from  (1),              x  —  y  = 

3; 

whence,                                                       y  = 

a: -3.                 *                (3) 

Substituting  (3)  in  (1) ,      x(ix-S)+x  = 

36, 

or                                                      x^  —  2x  = 

36. 

Solving,                                                   X  = 

7  or  -  5.                            (4) 

Substituting  (4)  in  (3),                        y  = 

4  or  -  8. 

Solve,  using  the  methods  illustrated 

in  exercises  1-5 ;  verify : 

(  m^-\-  mn  =  2, 
I  mw-f  7r  =  —1. 

xy  +  x  =  Z2, 

.xy  +  y  =  27. 

8. 

'a2-f&2  =  l30,                        ^^ 
.a-6  =  2. 

2x^-3^  =  5, 

SIMULTANEOUS   QUADRATIC   EQUATIONS         2T1 

372.  All  the  solutions  in  §§  362-371  are  but  illustrations  of 
methods  that  are  important  because  they  are  often  applicable. 
The  student  is  urged  to  use  his  ingenuity  in  devising  other 
methods  or  modifications  of  these  whenever  the  given  system 
does  not  yield  readily  to  the  devices  illustrated,  or  whenever 
a  simpler  solution  would  result. 


1. 


3. 


5. 


6. 


9. 


10. 


\^  +  f  = 


x^—  xy  =  48, 
xy  —  y-  =  12. 

r  a  +  aft  +  28  =  0, 

a^=-12. 

rr^  +  2/2  =  40, 
[xy=^12. 

fa.-»-f2/3  =  28, 
1  a;  +  2/  =  4. 

ra^-|-ar^  =  44, 

\xy  +  f=-2%.      ^ 

ric2  +  3ic.y-/  =  9, 
U  +  22/  =  4. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


MISCELLANEOUS  EXERCISES 

373.    Solve,  and  verify  each  result : 
^  +  2/  =  3, 
xy  =  2. 

5a^-42/'  =  44, 
4iB2-52/2  =  19. 

l  +  x  = 

61 


4if2/  =  96-ary, 
I  a;  +  2/  =  6. 

(Qi?  —  xy  =  ^f 
\xy  +  f=n. 

(x{x-\-y)  =  x, 

■x^  +  xy  +  f  =  l^l, 
.a^-f  2/2  =  106. 

(1+x^y, 
l4  +  4ar'  =  2/». 

(x^-y*  =  369, 
|a-2_2/2  =  9. 

x^  —  .Ty  =  6, 
i^r^  +  2/-  =  61. 

jic  +  2/  =  25, 


19 


^ 


^  +  xy-\-f=.l% 


xf 


=  19. 


20. 


'^  +  3xy  =  f^% 
a;  +  3  2/  =  5. 


272         SIMULTANEOUS   QUADRATIC   EQUATIONS 

Problems 

374.  1.  The  sum  of  two  numbers  is  12,  and  their  product  is 
32.     What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  17,  and  the  sum  of  their 
squares  is  157.     What  are  the  numbers  ? 

3.  The  difference  of  two  numbers  is  1,  and  the  difference  of 
their  cubes  is  91.     What  are  the  numbers  ? 

4.  The  area  of  a  rectangular  mirror  is  88  square  feet  and 
its  perimeter  is  38  feet.     Find  its  dimensions. 

5.  The  perimeter  of  a  rectangular  ginseng  bed  is  18  rods  and 
its  area  is  20J  square  rods.     What  are  its  demensions  ? 

6.  It  takes  52  rods  of  fence  to  inclose  a  rectangular  garden 
containing  1  acre.     How  long  and  how  wide  is  the  garden  ? 

7.  The  product  of  two  numbers  is  59  greater  than  their  sum, 
and  the  sum  of  their  squares  is  170.     What  are  the  numbers  ? 

8.  If  63  is  subtracted  from  a  certain  number  expressed  by 
two  digits,  its  digits  will  be  transposed ;  and  if  the  number  is 
multiplied  by  the  sum  of  its  digits,  the  product  will  be  729. 
What  is  the  number  ? 

9.  The  smallest  of  the  printed  muslin  flags  made  in  this 
country  has  an  area  of  5^  square  inches.  If  the  width  were  \ 
of  an  inch  less,  the  length  would  be  twice  the  width.  Find  the 
length  and  width. 

10.  A  man's  ice  bill  was  $  18.  If  ice  had  cost  $  1  less  per  ton, 
he  would  have  received  f  of  a  ton  more  for  the  same  money. 
How  many  tons  did  he  use  and  what  was  the  price  per  ton  ? 

11.  The  car  of  the  airship  America  is  107  feet  longer  than  it 
is  wide.  If  its  length  were  ^  as  great,  the  area  of  the  floor 
would  be  184  square  feet.     Find  its  dimensions. 

12.  A  Chinaman  received  $  1.60  for  rolling  joss  sticks. 
Had  he  been  paid  8  cents  more  for  each  lot  of  ten  thousand, 
he  would  have  had  to  roll  one  lot  less  for  the  same  amount  of 
money.  How  many  such  lots  did  he  roll  and  how  much  was 
he  paid  per  lot  ? 


SIMULTANEOUS   QUADRATIC   EQUATIONS         273 

13.  The  area  of  a  wall  painting  in  a  restaurant  in  Philadelphia 
is  340  square  feet.  If  its  length  were  12  feet  less  and  its  width 
12  feet  greater,  it  would  be  square.  Find  its  length  and 
width. 

14.  Before  the  reduction  in  letter  postage  between  the 
United  States  and  Great  Britain,  it  cost  10  ^  to  send  a  certain 
letter  that  now  would  just  go  for  4^.  What  is  the  weight  of 
the  letter,  if  the  postage  was  reduced  3  ^  an  ounce  ? 

15.  If  each  of  a  farmer's  maple  trees  had  yielded  2 
pounds  more  of  sugar,  he  could  have  made  750  pounds.  If  he 
had  had  50  trees  more,  he  could  have  made  600  pounds.  Find 
the  number  of  trees  he  had  and  the  yield  per  tree. 

16.  A  woman  in  Saxony  received  1^  an  hour  more  for 
making  chiffon  hats  than  for  weaving  straw  hats.  If  she  re- 
ceived 21  ^  for  her  work  on  the  former  and  20^  for  her  work 
on  the  latter,  working  14  hours  in  all,  how  much  did  she  re- 
ceive per  hour  for  each  ? 

17.  An  Illinois  farmer  raised  broom  corn  and  pressed  the 
6120  pounds  of  brush  into  bales.  If  he  had  made  each  bale  20 
pounds  heavier,  he  would  have  had  1  bale  less.  How  many 
bales  did  he  press  and  what  was  the  weight  of  each  ? 

18.  If  the  length  of  the  platform  of  an  elevator  were  9  feet 
less  and  its  width  9  feet  more,  its  area  would  be  361  square  feet. 
If  its  length  were  9  feet  more  and  its  width  9  feet  less,  its  area 
would  be  37  square  feet.     Find  its  dimensions. 

19.  A  man  bought  4  more  loads  of  sand  than  of  gravel,  pay- 
ing $.50  less  per  load  for  sand  than  for  gravel.  The  sand  cost 
him  $  9  and  the  gravel  $  10.  What  quantities  of  each  did 
he  buy  ?     What  prices  did  he  pay  ? 

20.  The  capital  stock  of  a  creamery  company  is  $  8850.  If 
there  had  been  10  times  as  many  stockholders,  each  man's 
share  would  have  been  $45  less.  How  many  stockholders 
were  there  and  what  was  each  man's  share  ? 

MILNE'S    IST    YR.     ALG.  18 


274         SIMULTANEOUS   QUADRATIC   EQUATIONS 

21.  Mr.  Fuller  paid  $2.25  for  some  Italian  olive  oil,  and 
$2  for  -J  gallon  less  of  French  olive  oil,  which  cost  $.50 
more  per  gallon.  How  much  of  each  kind  did  he  buy  and  at 
what  price  ? 

22.  In  papering  a  room,  18  yards  of  border  were  required, 
while  40  yards  of  paper  ^  yard  wide  were  needed  to  cover  the 
ceiling  exactly.     Find  the  length  and  breadth  of  the  room. 

23.  The  water  surface  area  of  a  tank  at  AVashington,  D.  C, 
used  in  testing  ship  models  is  20,210  square  feet.  If  the 
length  were  3  feet  greater,  it  would  be  11  times  the  width. 
What  are  the  dimensions  of  the  water  surface  ? 

24.  If  the  elm  beetle  had  killed  500  trees  less  one  year  in 
Albany,  New  York,  the  total  estimated  loss  would  have  been 
$  10,000 ;  if  the  value  of  each  tree  had  been  ^  as  much,  the 
total  loss  would  have  been  $  3750.  How  many  elm  trees 
were  killed  ? 

25.  The  total  area  of  a  window  screen  whose  length  is  4 
inches  greater  than  its  width  is  10  square  feet.  The  area 
inside  the  wooden  frame  is  8  square  feet.  Find  the  width  of 
the  frame. 

26.  A  rectangular  skating  rink  together  with  a  platform 
around  it  25  feet  wide  covered  37,500  square  feet  of  ground. 
The  area  of  the  platform  was  ^  of  the  area  of  the  rink.  What 
were  the  dimensions  of  the  rink? 

27.  The  course  for  a  36-mile  yacht  race  is  the  perimeter  of  a 
right  triangle,  one  leg  of  which  is  3  miles  longer  than  the  other. 
How  long  is  each  side  of  the  course  ? 

28.  At  simple  interest  a  sum  of  money  amounted  to  $2472 
in  9  months  and  to  $  2528  in  16  months.  Find  the  amount  of 
money  at  interest  and  the  rate. 

29.  Two  men  working  together  can  complete  a  piece  of  work 
in  6|  days.  If  it  would  take  one  man  3  days  longer  than  the 
other  to  do  the  work  alone,  in  how  many  days  can  each  man  do 
the  work  alone  ? 


GRAPHIC   SOLUTIONS 


QUADRATIC   EQUATIONS  —  TWO    UNKNOWN   NUMBERS 


375.    1.    Construct  the  graph  of  the  equation  a?-\-y'^  =  25. 

Solution.  —  Solving  for  y,  y=  ±  \/26  —  a;-^. 

Since  any  value  numerically  greater  than  5  substituted  for  x  will  make 
the  value  of  y  imaginary,  we  substitute  only  values  of  x  between  and  in- 
cluding —  5  and  -f  5.  The  corresponding  values  of  y,  or  of  ±  V25— x^, 
are  recorded  in  the  table  below. 

It  will  be  observed  that  each  value  substituted  for  x,  except  ±  5,  gives 
two  values  of  y,  and  that  values  of  x  numerically  equal  give  the  same 
values  of  y ;  thus,  when  x  =  2,  y  =  ±  4.6,  and  also  when  x  =  —  2, 
y=±  4.6. 


X 

y 

0 

±o 

±  1 

±4.9 

±  2 

±4.6 

±3 

±4 

±4 

±3 

±  5 

0 

■^ 

n 

"~ 

"" 

i 

J^'* 

A 

u* 

1 

^ 

* 

A 

Si 

/\ 

/ 

^ 

' 

V 

\ 

K 

/ 

jS 

L 

1 

i-* 

r 

_i- 

The  values  given  in  the  table  serve  to  locate  twenty  points  of  the  graph 
of  a;2  _|_  y2  _  25.  Plotting  these  points  and  drawing  a  smooth  curve 
through  them,  the  graph  is  apparently  a  circle.  It  may  be  proved  by 
geometry  that  this  graph  is  a  circle  whose  radius  is  5. 

The  graph  of  any  equation  of  the  form  jr^  4-/-  =  rMs  a  circle 
whose  radius  is  r  and.  whose  center  is  at  the  origin. 

276 


2T6 


GRAPHIC   SOLUTIONS 


2.   Construct  the  graph,  of  the  equation  y^  =  3  x  -\-  9. 

Solution.  —  Solving  for  y,  y  =  ±  VS  x  +  9, 

It  will  be  observed  that  any  value  smaller  than  —  3  substituted  for  x 
will  make  y  imaginary  ;  consequently,  no  point  of  the  graph  lies  to  the 
left  of  X  =  —  3.  Beginning  with  x  =  —  S,  we  substitute  values  for  x  and 
determine  the  corresponding  values  of  y,  as  recorded  in  the  table  : 


X 

y 

-3 

0 

-2 

±1.7 

-1 

±2.4 

0 

±3 

1 

±3.5 

2 

±3.9 

3 

±4.2 

-— $K'— 

===  ^    -  - 

--^- — ^- 

$ 

:iii±giiii: 

Plotting  these  points  and  drawing  a  smooth  curve  through  them,  the 
graph  obtained  is  apparently  a.  parabola  (§  356). 

The  graph  of  any  equation  of  the  form  y^  =  ax  -{■  c  is  a 
parabola. 

3.   Construct  the  graph  of  the  equation  9  a;^  +  25 1/^  =  225. 


Solution.  —  Solving  for  y,  y  =  ±  z  ^25  —  x^. 

Since  any  value  numerically  greater  than  5  substituted  for  x  will  make 
the  value  of  y  imaginary,  no  point  of  the  graph  lies  farther  to  the  right 
or  to  the  left  of  the  origin  than  5  units  ;  consequently,  we  substitute  for  x 
only  values  between  and  including  —  5  and  +  5. 

Corresponding  values  of  x  and  y  are  given  in  the  table : 


X 

y 

- 

0 

±1 
±2 
±3 

±4 

±5 

±3 
±2.9 
±2.7 
±2.4 
±  1.8 
0 

p* 

r-t 

L^-^''^'- 

rH 

^ 

if^^^ 

^^1 

/ 

?r 

? 

s 

^ 

\ 

f 

V 

1^ 

H 

*-!< 

^ 

^ 

r 

^ 

J 

_ 

_ 

_ 

_ 



GRAPHIC   SOLUTIONS 


277 


Plotting  the  twenty  points  tabulated  on  the  preceding  page,  and  draw- 
ing a  smooth  curve  through  them,  we  have  the  graph  of  Qx^  +  2oy^ 
=  225,  which  is  called  an  ellipse. 

.    The  graph  of  any  equation  of  the  form  b^x^  +  a^^  =  a^b^  is  an 
ellipse. 

4.    Construct  the  graph  of  the  equation  4:X'—9y^  =  36. 

SoLUTioy 

Solving  f or  y,  y  =  ±  §  Vx"'^  —  9. 

Since  any  value  numerically  less  than  3  substituted  for  x  will  make 
the  value  of  y  imaginary,  no  point  of  the  graph  lies  between  x  =  +  3  and 
X  =  —  3 ;  consequentlj^,  we  substitute  for  x  only  ±  3  and  values  nnraer- 
ically  greater  than  3. 

Corresponding  values  of  x  and  y  are  given  in  the  table : 


X 

y 

>^ 

k 

/ 

Y'^i 

- 

> 

k 

> 

^^ 

±3 
±4 

±6 
±6 

0 
±1.8 

±2.7 
±  3.5 

T 

w 

i 

J^ 

* 

\ 

i 

K^' 

\ 

L. 

J 

Yi 

Vs 

1 

_ 

Plotting  these  fourteen  points,  it  is  found  that  half  of  them  are  on  one 
side  of  the  j/-axis  and  half  on  the  other  side,  and  since  there  are  no  points 
of  the  curve  between  x  =  +  3  and  x  =  —  3,  the  graph  has  two  separate 
branches^  that  is,  it  is  discontinuous. 

Drawing  a  smooth  curve  through  each  group  of  points,  the  two  branches 
thus  constructed  constitute  the  graph  of  the  equation  4  x^  —  9  y2  =  36, 
which  is  an  hijperhola. 

The  graph  of  any  equation  of  the  form  b'x^  —  ary-  =  arb"^  is  an 
hyperbola. 

An  hyperbola  has  two  branches  and  is  called  a  discontinuous 
curve. 


278 


GRAPHIC   SOLUTIONS 


6.   Construct  the  graph  of  the  equation  xy  =  10. 
Solution.  —  Substituting  values  for  x  and  solving  for  y,  the  corre- 
sponding values  found  are  as  given  in  the  table  : 


X 

y 

1 

X 

y 

1 

10 

-1 

-10 

2 

.5 

-2 

-5 

3 

3^ 

-3 

-^ 

4 

n 

-4 

-n 

5 

2 

-5 

-2 

6 

1| 

-6 

-It 

7* 

If 

-7 

-H 

8 

H 

-8 

-H 

9 

H 

-9 

-H 

10 

1 

-  10 

-  1 

— 

~ 

P 

— 

— 

— 

— 

— 

\ 

\ 

\ 

V^^ 

N 

^  * 

/^ 

V 

N 

^ 

&-* 

. 

^ 

^ 

L, 

> 

K\ 

A 

^ 

_ 

_ 

_ 

_ 

^ 

_ 

_ 

Plotting  these  points  and  drawing  a  smooth  curve  through  each  group 
of  points,  the  two  branches  of  the  curve  found  constitute  the  graph  of  the 
equation  xy  —  10,  which  is  an  hyperbola. 

The  graph  of  any  equation  of  the  form  jr/  =  <?  is  an  hyperbola. 
Construct  the  graph  of : 

6.  ic2  +  /  =  9.  8.    9  a^  +  16/  =  144.  10.    xy  =  12. 

7.  2/' =  5  a;  4- 8.         9.    9  a;^  - 16  /  =  144.       '  11.   xy  =  -  &, 

376.  Summary.  —  The  types  of  equations  and  their  respec- 
tive graphs,  here  given,  vrill  aid  the  student  in  plotting  graphs, 
but  he  will  meet  other  forms  of  equations  that  will  have  some 
of  the  same  kinds  of  graphs,  the  varieties  in  equations  giving 
rise  to  varieties  in  form,  size,  or  location  of  the  graphs. 

For  example,  §  375,  exercises  4  and  5  are  both  equations  of  the  hyper- 
bola, but  they  are  differently  located  and  of  different  size  and  shape. 

It  is  possible  to  determine  many  characteristics  of  the  vari- 
ous graphs  from  their  equations  alone,  but  a  discussion  of  this 
is  beyond  the  province  of  algebra.  In  the  study  of  graphs, 
therefore,  the  student  will  rely  principally  on  plotting  a  suffi- 


GRAPHIC   SOLUTIONS 


279 


(§356) 


Straight  line 
Circle 

Parabola 


cient  number  of  points  to  determine  their  form  accurately.     The 
following  types  have  been  studied  : 
I.  fljr  +  6/ =  c  (§  248) 
II.   jr2+/  =  /^ 

III      /  ^^^  "*"  ^-'^  "f"  ^  =  Oj  or 

\y  z=  ax- -\- bx -\- c 

IV.  y"^  =  ax  +  c  Parabola 

V.   6V  +  ay  =  a'62  Ellipse 

VI.   b'x' -  ay  =  a^'  Hyperbola 

VII.    xy  =  c  Hyperbola 

SIMULTANEOUS  QUADRATIC  EQUATIONS 

377.  The  graphic  method  of  solving  simultaneous  equations 
that  involve  quadratics  is  precisely  the  same  as  for  simulta- 
neous linear  equations  (§  252).  Construct  the  graph  of  each 
equation,  both  graphs  being  referred  to  the  same  axes,  and  de- 
termine the  coordinates  of  the  points  where  the  graphs  inter- 
sect.    If  they  do  not  intersect,  interpret  this  fact. 

Note.  — The  student  should  construct  the  following  graphs  for  himself. 
Roots  are  expected  to  the  nearest  tenth  of  a  unit.  To  obtain  this  degree 
of  accuracy,  numerous  points  should  be  plotted  and  a  scale  of  about  \  inch 
to  1  unit  should  be  used. 


EXERCISES 


378.     1.   Solve  graphically  I  ^JI"  ^^-_^^' 


Solution.  —  Constructing  the 
graphs  of  these  equations,  we  find 
the  first,  as  in  §375,  exercise  1,  to  be 
a  circle  ;  and  the  second,  as  in  §  248, 
a  straight  line. 

The  line  intersects  the  circle  in  two 
points  (—4,  -  3)  and  (3,  4). 

Hence,   there  are  two  solutions 

a;=-4,  y  =  -3  and  a;  =  3,  y  =  4. 

Test.  —  The  student  may  test  the 
roots  found  graphically  by  performing 
the  algebraic  solution. 


f_ 

Z 

d^"" 

^^^ 

-fey 

^^r- 

-U 

Z      ^ 

IE 

z       _^ 

-dt    -.z 

i  ^ 

v^ 

1 

%t 

T 

2^^ 

z 

7       "^^^ 

^^ 

280 


GRAPHIC   SOLUTIONS 


2.    Solve  graphically 


y  =  4 

. 

4^ 

■""■ 

y^2 

1"'' 

n^ 

V 

-\ 

■ 

^. 

/ 

y 

__.— * 

'^\ 

b:- 

9.X^ 

_L 

3.    Solve  graphically 


9a;2  +  252/'  =  225, 

Solution. — On  constructing  the 
■graphs  (for  the  first,  see  exercise  3, 
§  375),  it  is  found  that  they  intersect 
at  the  points  x  =  3.7,  y  ■=^  and  x  = 
-3.7,^  =  2. 

Since  the  graphs  have  these  two 
points  in  common,  and  no  others,  their 
coordinates  are  the  only  values  of  x  and 
y  that  satisfy  both  equations  and  are 
the  roots  sought. 

Observe  that  the  pairs  of  values  x  = 
3.7,  y  =  2    and    a;=  — 3.7,?/  =  2   are 
reaZ,  and  different,  or  unequal. 
Note.  —  The  roots  are  estimated  to  the  nearest  tenth  ;  their  accuracy 
may  be  tested  by  performing  the  algebraic  solution. 

9a;2  +  252/'  =  225, 

Solution.— Imagine  the  straight  line  ?/  =  2  in  the  figure  for  exercise  2 
to  move  upward  until  it  coincides  with  the  line  y  —  3.  The  real  unequal 
roots  represented  by  the  coordinates  of  the  points  of  intersection  approach 
equality,  and  when  the  line  becomes  the  tangent  line  y  =  3,  they  coincide. 
Hence,  the  given  system  of  equations  has  two  real  equal  roots  ic  =  0, 
?/  =  3  and  x  =  0,  i/  =  3. 

935^+25  2/2=225, 
2/ =  4. 

Solution. — Imafjine  the  straight  line  y  =  2  in  the  figure  for  exercise  2 
to  move  upward  until  it  coincides  with  the  line  ?/  =  4.  The  graphs  will 
cease  to  have  any  points  in  common,  showing  that  the  given  equations 
have  no  common  real  values  of  x  and  y. 

It  is  shown  by  the  algebraic  solution  of  the  equations  that  there  are 
two  roots  and  that  both  are  imaginary. 

A  system  of  two  independent  simultaneous  equations  in  x  and  y, 
one  simple  and  the  other  quadratic,  has  two  roots. 

TJie  roots  are  real  and  unequal  if  the  graphs  intersect,  real  and 
equal  if  the  graphs  are  tangent  to  each  other,  and  imaginary  if  the 
graphs  have  no  points  in  common. 


4.   Find  the  nature  of  the  roots  of 


GRAPHIC   SOLUTIONS 


281 


■#i.^  — 

'^v  iy\  !^^*e       \  ^^-i** 

t^        ^^ 

tsx       ::i^ji 

v''*^"            '^■■-        -'^               ^Nv. 

^^^^           ^Z    ^s^ 

5.  Solve  graphically 

Solution. — The  graphs  (see  ex- 
ercises 4  and  1,  §  375)  show  that 
both  of  the  given  equations  are  satis- 
fied by  four  different  j^airs  of  real 
values  of  x  and  y  : 

fx=4.5;       4.5;  -4.5;  -4.6; 

[y  =  2.2;  -2.2;  -2.2;       2.2. 

6.  What  would  be  the  nature  of  the  roots  in  exercise  5,  if 
the  second  equation  were  ar^  +  2/^  =  9  ? 

Solution,  —  Imagine  the  circle  a;^ -f- y2  =  25  in  exercise  5  to  become 
smaller  and  smaller  until  it  coincides  with  the  circle  x^  +  yS  —  g  (gge 
dotted  circle  in  the  cut) .  The  four  real  unequal  roots  represented  by  the 
coordinates  of  the  points  of  intersection  of  the  graphs  come  together  in 
pairs  at  the  points  (3,  0)  and  (—3,  0)  where  the  circle  x'^  +  y'^  =  9  is 
tangent  to  the  hyperbola  4  a;^  _  9  y2  _  35  .  consequently,  the  equations 
4  x2  —  9  j/2  =  36  and  x^  -f-  y*  =  9  have  two  pairs  of  equal  real  roots,  namely : 
fx  =  3,  3,  -3,-3; 
ly  =  0,0,      0,      0. 

A  system  of  two  independent  simultaneous  quadratic  equations 
in  X  and  y  has  four  roots. 

An  intersection  of  the  graphs  represents  a  real  root,  and  a 
point  of  tangency,  a  pair  of  equal  real  roots.  If  there  are  less 
than  four  real  roots,  the  other  roots  are  imaginary. 

It  is  not  possible  to  solve  any  two  simultaneous  equations 
in  X  and  y,  that  involve  quadratics,  by  quadratic  methods, 
but  approximate  values  of  the  real  roots  may  always  be  found 
by  the  graphic  method. 

Find  by  graphic  methods,  to  the  nearest  tenth,  the  real  roots 
of  the  following,  and  the  number  of  imaginary  roots,  if  there 
are  any.     Discuss  the  graphs  and  the  roots. 

'4ar^-92/^  =  36,  ^      {4.x'-9y'=S6, 


8. 


x-Sy  =  l. 

4a;2-9/  =  36, 
4  a;2  4. 9  2/2  =  36. 


9.     \ 

U2/  =  a^-16. 

jQ      r9a^  +  16y^  =  144, 

[Sx-^4:y^l2. 


282  GRAPHIC   SOLUTIONS 

,2.    p^  +  2/^  =  4,  ^^     ix^  +  f  =  % 


6. 


x  =  y-5.  ^x  =  y^  +  5y-h6, 

13      f^-^2/'  =  4,  ^^      fy  =  x^-4., 


a;?/  =  —  1-  I  a;  =  2/^  —  4  2/  +  3. 

379.  Another  graphic  method  of  solving  quadratic  equations  in 
one  unknown  number  (§  356). 

It  has  been  seen  that  the  real  roots  of  simultaneous  equations 
are  the  coordinates  of  the  points  where  their  graphs  intersect 
or  are  tangent  to  each  other,  and  that  when  there  is  no  point 
in  common,  the  roots  are  imaginary. 

In  §§  356,  357,  it  was  found  that  the  real  roots  of  a  quadratic 
equation  were  the  abscissas  of  the  points  where  the  graph  of 
the  quadratic  expression  crossed  or  touched  the  x-axis,  and 
that  when  it  had  no  point  in  common  with  the  aj-axis,  the  roots 
were  imaginary. 

In  other  words,  the  solution  of  a  quadratic  equation  in  x  was 
made  to  depend  upon  the  solution  of  the  simultaneous  system, 

,-|-.  f  2/  =  ^  -\-hx-\-Cf  (a  parabola) 

\y  =  0,  (a  straight  line) 

the  second  being  the  equation  of  the  aj-axis. 

In  the  following  method,  by  substituting  y  for  x'^  in  the  given 
equation, 

ax^  4-  &a?  +  c  =  0, 

the  equation  is  divided  into  the  simultaneous  system, 

,-j-yv  {ay  +  hx-\-c=Oj  (a  straight  line) 

\yz=a?.  (a  parabola) 

The  solution  of  this  system  for  x  gives  the  required  roots  of 
aa?  +  &a;  +  c  =  0. 


GRAPHIC   SOLUTIONS 


288 


It  will  be  observed  that  whether  system  (I)  or  (II)  is  used, 
the  solution  requires  the  construction  of  a  parabola  and  a 
straight  line,  but  the  advantage  of  using  (II)  instead  of  (I)  lies 
in  the  fact  that  the  parabola  y  =  x^  is  the  same  for  all  quadratic 
equations  in  x  and  when  once  constructed  can  be  used  for  solv- 
ing any  number  of  equations,  while  with  (I)  a  different  parabola 
must  be  constructed  for  each  equation  solved. 


EXERCISES 

380.     1.    Solve  graphically  the  equation  x^  —  2  a;  —  8  =  0. 

Solution 
Substituting  y  for  x^,  we  have 

y-2x-8  =  0. 
Consequently,  the  values  of  x 
that  satisfy  the  system, 
ly-2x-S  =  0, 

are  the  same  as  those  that  satisfy 
the  given  equation. 

Constructing  the  graph  of 
y  =  x2,  we  have  the  parabola 
shown  in  the  figure. 

Constructing  the  graph  of 
y  —  2x  —  S  =  0,?i  straight  line,  we 
find  that  it  intersects  the  parabola 
at  a;  =  —2  and  x  =  4. 

Hence,  the  roots  of  the  equation 
a;2  -  2  a;  -  8  =  0  are  —  2  and  4. 

->Solve  graphically,  giving  roots  to  the  nearest  tenth: 

2.  3.2  +  3.-2  =  0.  8.  2x'-x  =  6. 

3.  x^~x-6  =  0.  9.  2a^-x  —  15  =  0. 

4.  3c^-Sx-4:  =  0.  10.  3aj2-|-5a;-28  =  0. 

5.  x'-2x-W  =  0:  11.  6x^-7x-20  =  0. 

6.  x'-\-5x-{-U  =  0.  12.  Sx^-\-Ux-15  =  0. 

7.  a;2_7a;-f.l8  =  0.  13.  15 ar^ -j- 2 a; - 20  =  0. 


RATIO  AND  PROPORTION 


RATIO 


381.  The  relation  of  two  numbers  that  is  expressed  by  the 
quotient  of  the  first  divided  by  the  second  is  called  their  ratio. 

382.  The  sign  of  ratio  is  a  colon  (:). 

A  ratio  is  expressed  also  in  the  form  of  a  fraction. 

The  ratio  of  a  to  6  is  written  a  ;  6  or  ^  • 

0 

The  colon  is  sometimes  regarded  as  derived  from  the  sign  of  division 
by  omitting  the  hne. 

383.  To  compare  two  quantities  they  must  be  expressed  in 
terms  of  a  common  unit. 

Thus,  to  indicate  the  ratio  of  20^  to  $1,  both  quantities  must  be  ex- 
pressed either  in  cents  or  in  dollars,  as  20^  :  100^  or  $  ^ :  $  1. 
There  can  be  no  ratio  between  2  pounds  and  3  feet. 

The  ratio  of  two  quantities  is  the  ratio  of  their  numerical 
measures. 

Thus,  the  ratio  of  4  rods  to  5  rods  is  the  ratio  of  4  to  5. 

384.  The  first  term  of  a  ratio  is  called  the  antecedent,  and 
the  second,  the  consequent.     Both  terms  form  a  couplet. 

The  antecedent  corresponds  to  a  dividend  or  numerator ;  the 
consequent,  to  a  divisor  or  denominator.  . 

In  the  ratio  a  :  6,  or  -  ,  a  is  the  antecedent,  h  the  consequent,  and 
6 

the  terms  a  and  h  form  a  couplet. 

284 


RATIO  AND  PROPORTION  285 

385.  The  ratio  of  the  reciprocals  of  two  numbers  is  called 
the  reciprocal,  or  inverse,  ratio  of  the  numbers. 

It  may  be  expressed  by  interchanging  the  terms  of  the 
couplet. 

The  inverse  ratio  of  a  to  &  is  -  :  -•    Since  -  —  -  =  -,  the  inverse  ratio 
a    b  aba 

of  a  to  b  may  be  written  -  or  &  :  a. 
a 

Properties  of  Ratios 

■  386.  It  is  evident  from  the  definition  of  a  ratio  that  ratios 
have  the  same  properties  as  fractions ;  that  is,  they  may  be 
reduced  to  higher  or  lower  ter7ns,  added,  subtracted,  etc.  Hence, 
Principles.  —  1.  Multiplying  or  dividing  both  terms  of  a 
ratio  by  the  same  number  does  not  change  the  value  of  the  ratio. 

2.  Mvltiplying  the  antecedent  or  dividing  the  consequent  of  a 
ratio  by  any  number  multiplies  the  ratio  by  that  number. 

3.  Dividing  the  antecedent  or  multiplying  the  consequent  by 
any  number  divides  the  ratio  by  that  number. 

EXERCISES 

387.     1.   What  is  the  ratio  of  8  m  to  4  m  ?  of  4  m  to  8  m  ? 

2.  Express  the  ratio  of  6 : 9  in  its  lowest  terms ;  the  ratio 
12x:16y;  amibm-,  20a6:  106c;  (m  i- n)  :  (m^  -  n^. 

3.  Which  is  the  greater  ratio,  2:3  or  3:4?  4:9  or  2:5? 

4.  What  is  the  ratio  of  ^  to  J  ?  |  to  |  ?  |  to  |  ? 

Suggestion.  —  When  fractions  have  a  common  denominator,  they 
have  the  ratio  of  their  numerat6rs. 

5.  What  is  the  inverse  ratio  of  3  :  10  ?  of  12  :  7  ? 
Reduce  to  lowest  terms  the  ratios  expressed  by : 

6.  10:2.  8.    3:27.  10.    ^.  12.    75-100. 

7.  12:6.  9.   4:40.  11.    ^f.  13.    60-120. 

14.  What  is  the  ratio  of  15  days  to  30  days  ?  of  21  days  to 
1  week  ?  of  1  rod  to  1  mile  ? 


286  RATIO   AND  PROPORTION 

Find  the  value  of  each  of  the  following  ratios : 

15.  ixi^x".        17.    21 :  7-1         19.    a'b'x' :  a^6V. 

16.  }a6:|ac.     18.    .7m:  .Sn.     20.    (q^ -y^)  :  (x  —  yf. 

21.  If  9  is  subtracted  from  4  and  then  from  5,  find  the  ratio 
of  the  first  remainder  to  the  second. 

22.  Change  each  to  a  ratio  whose  antecedent  shall  be  1  : 

5:20;    3a;:12a;;    f:f;    .4:1.2. 

23.  When  the  antecedent  is  6  a;  and  the  ratio  is  I,  what  is 
the  consequent? 

PROPORTION 

388.  An  equality  of  ratios  is  called  a  proportion. 

3  :  10  =  6  :  20  and  a:x  =  b:y  are  proportions. 

The  double  colon  (::)  is  often  used  instead  of  the  sign  of 
equality. 

The  double  colon  has  been  supposed  to  represent  the  extremities  of  the 
lines  that  form  the  si^  of  equality. 

The  proportion  a:b  =  c:dj  or  a:b::c:d,  is  read,  ' the  ratio 
of  a  to  &  is  equal  to  the  ratio  of  c  to  d,'  or  ^  a  is  to  6  as  c  is  to  d' 

389.  In  a  proportion,  the  first  and  fourth  terms  are  called 
the  extremes,  and  the  second  and  third  terms,  the  means. 

In  a  :b  =  c:d,  a  and  d  are  the  extremes,  b  and  c  are  the  means. 

390.  Since  a  proportion  is  an  equality  of  ratios  each  of 
which  may  be  expressed  as  a  fraction,  a  proportion  may  be 
expressed  as  an  equation  each  member  of  which  is  a  fraction. 

Hence,  it  follows  that : 

General  Principle.  —  The  changes  that  may  be  made  in  a 
proportion  without  destroyiyig  the  equality  of  its  ratios  correspond 
to  the  changes  that  may  be  made  in  the  members  of  an  equation 
without  destroying  their  equality  and  in  the  terms  of  a  fraction 
ivithout  altering  the  vahie  of  the  fraction. 


RATIO  AND  PROPORTION  287 

Properties  of  Proportions 

391.  Pkixciple  1.  —  In  any  pi'oporticm  the  product  of  the  ex- 
tremes is  equal  to  the  product  of  the  means. 

For,  given  a:b  =  c:d, 

a     c 

Clearing  of  fractions,  ad  =  be. 

Test  the  following  by  principle  1  to  find  whether  they  are 
true  proportions: 

1.    6:16  =  3:8.  2.    ||  =  if-  3.    7:8=10:12 

392.  In  the  proportion  a:m  =  m:b,  m  is  called  a  mean  pro- 
portional between  a  and  b. 

By  Prin.  1,  m^  =  ab; 

.'.  m  =  Va6. 

Hence,  a  mean  proportional  between  two  numbers  is  equal  to 
the  square  root  of  their  product. 

1.  Show  that  the  mean  proportional  between  3  and  12  is 
either  6  or  —  6.     Write  both  proportions. 

2.  Find  two  mean  proportionals  between  4  and  25. 

393.  Principle  2.  —  Either  extreme  of  a  proportion  is  equal 
to  the  product  of  the  means  divided  by  the  other  extreme. 

Either  mean  is  equal  to  the  product  of  the  extremes  divided  by 
the  other  mean. 

For,  given  a:b  =  c:d. 

By  Prin.  1,  ad  =  he. 

Solving  for  a,  d,  &,  and  c,  in  succession,  Ax.  4, 
he    ■,      he  ,      ad  ad 

d  a  c  h 

1.  Solve  the  proportion  3:4:  — x:  20,  for  x. 

2.  Solve  the  proportion  x:a  =  2m:n,  for  a;. 


288  KATIO  AND  PROPORTION 

3.  If  a  :  6  =  &  :  c,  the  term  c  is  called  a  third  proportional  to  a 
and  b.     Find,  a  third  proportional  to  6  and  2. 

4.  In  the  proportion  a:b  =  c-.d,  the  term  d  is  called  a  fourth 
proportional  to  a,  6,  and  c.     Find  a  fourth  proportional  to  J,  ^y 

and  -J. 

394.  Principle  3.  —  If  the  product  of  two  numbers  is  equal 
to  the  product  of  two  other  numbers,  one  pair  of  them  may  be 
made  the  extremes  and  the  other  pair  the  means  of  a  proportion. 

For,  given  ad=  be. 

Dividing  by  bd,  Ax.  4,  ^  =  ^ ; 

b     d 

that  is,  a  :b  =  c:d. 

By  dividing  both  members  of  the  given  equation,  or  of  be  =  ad,  by 
the  proper  numbers,  various  proportions  may  be  obtained  ;  but  in  all 
of  them  a  and  d  will  be  the  extremes  and  b  and  c  the  means,  or  vice 
versa,  as  illustrated  in  the  proofs  of  principles  4  and  5. 

1.  If  a  men  can  do  a  piece  of  work  in  x  days,  and  if  b  men 
can  do  the  same  work  in  y  days,  the  number  of  days'  work  for 
one  man  may  be  expressed  by  either  ax  or  by.  Form  a  pro- 
portion between  a,  b,  x,  and  y. 

2.  The  formula  pd  =  WD  (See  p.  166) 

expresses  the  physical  law  that,  when  a  lever  just  balances, 
the  product  of  the  numerical  measures  of  the  power  and  its 
distance  from  the  fulcrum  is  equal  to  the  product  of  the 
numerical  measures  of  the  weight  and  its  distance  from  the 
fulcrum.     Express  this  law  by  means  of  a  proportion. 

395.  PtiiNCiPLE  4.  —  If  four  numbers  are  in  2woportion,  the 
ratio  of  the  antecedents  is  equal  to  the  ratio  of  the  consequents; 
that  isj  the  numbers  are  in  proportion  by  alternation. 

For;  given  a:b  =  c:d. 

Then,  Prin.  1,  ad  =  bc. 

Dividing  by  cd,  Ax.  4,  ^  =  - ; 

c     d 

that  is,  a  :c=:b  :d. 


RATIO   AND  PROPORTION  289 

396.  Principle  5.  —  If  four  numbers  are  in  proportion,  the 
ratio  of  the  second  to  the  first  is  equal  to  the  ratio  of  the  fourth 
to  the  third;  that  is,  the  numbers  are  in  proportion  by  inversion. 


For,  given 

a:bz=c:d. 

Then,  Prin.  1, 

ad  =  bc. 

.'.  be  =  ad. 

Dividing  by  ac,  Ax.  4, 

b_d, 
a     c  ' 

that  is, 

b  :  a  =  d  :  c. 

397.  Principle  6.  —  If  four  numbers  are  in  proportion,  the 
sum  of  the  terms  of  the  first  ratio  is  to  either  term  of  the  first  ratio 
as  the  sum  of  the  terms  of  the  second  ratio  is  to  the  corresponding 
term  of  the  second  ratio  ;  that  is,  the  numbers  are  in  proportion 
by  composition. 

For,  given  a:b  =  c:d, 


Then,  Ax.  1, 


a 

c 

b' 

~d 

a 

+  1    : 

=^+l» 

b 

d 

a 

+  6. 

c  +  d 

b 

d     ' 

that  is,  a  +  b  :b  =  c-\-  d:d. 

Similarly,  taking  the  given  proportion  by  inversion  (Prin.  6),  and  add- 
ing 1  to  both  members,  we  obtain 

a-\-b  :a  =  c  +  d:c. 

393.  Principle  7.  —  If  four  numbers  are  in  propoHion,  the 
difference  between  the  terms  of  the  first  ratio  is  to  either  term  of 
the  first  ratio  as  the  difference  between  the  terms  of  the  second 
ratio  is  to  the  corresponding  term  of  the  second  ratio;  that  is, 
the  numbers  are  in  proportion  by  division. 

For,  in  the  proof  of  Prin.  6,  if  1  is  subtracted  instead  of  added,  the 
following  proportions  are  obtained : 

a  —  b  '.  h  ^=  c  —  d  '.  d^ 
^^^  a  —  b  :  a  =  c  ~  d  :  c. 

milne's  1st  yr.  alg.  — 19 


290  RATIO   AND   PROPORTION 

399.  Principle  8.  —  If  four  numbers  are  in  proportion,  the 
sum  of  the  terms  of  the  first  ratio  is  to  their  difference  as  the  sum 
of  the  terms  of  the  second  ratio  is  to  their  difference;  that  is,  the 
numbers  are  in  proportion  by  composition  and  division. 


For,  given 

a  :b  =  c  :d. 

By  Prin.  6, 

■  a  +  b  _cA-d 
b           d 

(1) 

By  Prin.  7, 

a  —  b  _c  —  d 
b            d 

(2) 

Dividing  (1)  by  (2), 

Ax. 

4, 

a  +  h  _c  +  d , 
a-h     c-d' 

at  is, 

a  +  b:a—b  =  ci-d:G- 

-d. 

400.  Principle  9.  —  In  a  proportion,  if  both  terms  of  a  couplet 
or  both  antecedents,  or  both  consequents  are  multiplied  or  divided 
by  the  same  number,  the  resulting  four  numbers  form  a  proportion. 

For,  given  a:b  =  C'.d, 


a  _  c 
b     d' 

mb     nd 

a 
b 

n      d     n 

Then,  §181, 

Also,  Ax.  3,  -•—  =  -'—,  or  ma  :  nb  =  mc  :  nd. 

b      n      d     n 

401.  Principle  10.  —  In  a  series  of  equal  ratios,  the  sum  of 
all  the  antecedents  is  to  the  sum  of  all  the  consequents  as  any 
antecedent  is  to  its  consequent. 


For,  given 

a:b  =  c:d  =  e:f, 

or 

?  =  J  =  i  =  r,theval 
b     d    f 

Then,  Ax.  3, 

a  =  br,  c  =  dr,  e  =fr 

whence.  Ax.  1, 

a-\-c  +  e={b  +  d+f)r, 

"  b  +  d-\-f           b     d     f 

that  is,  a  +  c  +  e:6  +  d+f=a  :  6  or  c  :  d  or  e  :/. 


RATIO  AND  PROPORTION  291 

EXERCISES 

402.     1.    Find  the  value  of  x  in  the  proportion  3  :  5  =  a;:  55. 

Solution.  3  :  5  =  x  :  66, 

Prin.  2,  a;  =  ^  =  33. 

5 

Find  the  value  of  x  in  each  of  the  following  proportions : 

2.  2  : 3  =  6  :  a;.  5.    .'c  +  2 :  a;  =  10  :  6. 

3.  5:  a;  =  4:  3.  6.    it- :  a;  -  1  =  15: 12. 

4.  l:a;  =  a;:9.  7.    a;  +  2:  a;  —  2  =  3:1. 

8.  Show  that  a  mean  proportional  between  any  two  num- 
bers having  like  signs  has  the  sign  ± . 

9.  Find  two  mean  proportionals  between  V2  and  V8. 

10.  Find  a  third  proportional  to  4  and  6. 

11.  Find  a  fourth  proportional  to  3,  8,  and  7^. 

Test  to  see  whether  the  following  are  true,  proportions : 

12.  5  J :  3  =  4  :  1^.  14.    5  :  7  a^  =  10 :  14  x. 

13.  4:13  =  2:6^  15.    2.4  a:  .8  a  =  6  a:  2  a. 

16.  Given  a  :  b  =c  :  d, 

to  prove  that     2a  +  3c:2a-3c  =  864-12d:86-12d 

Proof. — Given  a  :b  =  c:  d. 

By  alternation,  Prin.  4,  a:  c  =  b  :  d. 

Expressing  as  a  fractional  equation,     -  =  -  • 

c      d 

Multiplying  the  first  mepaber  by  f  and  the  second  by  ^,  the  equal  of  |, 

2a^ 8ft . 

3  c      12  d ' 
that  is,  2  a  :  3  c  =  8  6  :  12  d. 

By  composition  and  division,  Prin.  8, 

2a-{-Sc:2a-3c  =  Sb  + 12  d:Sb- 12  d. 
When  a:b  =  r:  d,  prove  that: 

17.  d:b  =  c:  a. 

18.  c:d  =  -:-. 

b   a 

19.  ¥  :  d^  =  if  :  c^ 


20. 

a' :  6  V  =  1 

:d?. 

21. 

b 
ma :  -  =  mc 

'2* 

22. 

ac:bd=(^: 

d\ 

292  RATIO   AND  PROPORTION 

When  a:b  =  c:d,  prove  that : 

23.  a -\- b:c -\-d  =  a —  b:c  —  d. 

24.  2a-i-  5b:2a  =  2c  +  5d:2c. 

25.  4a  -Sb:4:C-Sd  =  a:c. 

26.  a  :  a  -\-  b  =  a  -{-  c :  a  -\-  b  -\-c  -\-  d. 

27.  a-\-b:c  +  d=  Va'  +  6^:  V^T^- 

Problems 

403.  1.  The  ratio  of  the  rate  of  a  local  train  in  the  New  York 
subway  to  an  express  train  is  1 :  2.  If  the  local  train  runs  15 
miles  an  hour,  find  the  rate  of  the  express  train. 

2.  The  consumption  of  gas  in  New  York  City  one  year  was 
to  that  in  Chicago  as  7  : 4.  If  12  billion  cubic  feet  were  con- 
sumed in  Chicago,  what  was  the  consumption  in  New  York  ? 

3.  Michigan  produces  yearly  25  %  of  the  iron  ore  of  the 
United  States.  The  ratio  of  Michigan's  output  to  Minne- 
sota's is  5:8.  AVhat  per  cent  of  the  country's  output  does 
Minnesota  produce  ? 

4.  The  United  States  manufactured  285  million  pens  one 
year.  The  ratio  of  the  steel  pens  to  the  whole  number  of 
pens  was  18 :  19.     How  many  steel  pens  were  manufactured  ? 

5.  A  diver  descended  210  feet  into  a  lake.  The  ratio  of 
this  distance  to  the  distance  that  is  usually  considered  the 
limit  for  divers  is  7 :  5.     Find  the  usual  limit  for  divers. 

6.  How  many  pounds  of  tea  are  made  from  4200  pounds  of 
the  green  leaf,  if  the  ratio  of  the  weight  of  the  manufactured 
tea  to  that  of  the  green  leaf  is  5 :  21  ? 

7.  Two  machines,  one  old  and  one  modern,  turn  out  960  pins 
per  minute.  The  ratio  of  the  number  turned  out  by  the  old 
machine  to  the  number  turned  out  by  the  modern  one  is  1 :  15. 
How  many  were  turned  out  by  each  machine  ? 


RATIO  AND  PROPORTION  "  293 

8.  Find  a  number  that  added  to  each  of  the  numbers  1,  2, 
4,  and  7  will  give  four  numbers  in  proportion. 

9.  The  United  States  published  20,000  newspapers  recently. 
The  relation  of  this  number  to  those  published  in  the  whole 
world  was  2 : 5.     How  many  were  published  in  the  world  ? 

10.  The  ratio  of  the  greatest  length  of  Lake  Erie  to  the 
greatest  length  of  Lake  Michigan  is  5  :  6.  What' is  the  length 
of  each,  if  Lake  Michigan  is  50  miles  longer  than  Lake  Erie  ? 

11.  The  ratio  of  the  loss  of  life  in  the  Lisbon  earthquake  to 
that  in  the  Messina  earthquake  is  12  :  23.  If  55,000. more  lives 
were  lost  in  the  latter  than  in  the  former,  find  the  loss  of  life 
in  each  earthquake. 

12.  The  length  of  a  giant  candle  was  to  that  of  a  Christmas 
candle  as  40 :  1.  If  8  times  the  length  of  the  latter  was  96 
inches  less  than  that  of  the  former,  find  the  length  of  each. 

13.  The  wool  sales  for  one  week  in  New  York  amounted  to 
555,000  pounds.  The  ratio  of  the  domestic  sales  to  the  foreign 
was  14 :  23.     What  were  the  foreign  sales  ? 

14.  Out  of  a  lot  of  shell  caps,  100  times  the  number  rejected 
by  the  government  inspector  for  imperfections  was  to  the  total 
number  as  3  :  11.  If  1097  were  accepted,  how  many  were 
rejected  ? 

15.  In  one  year  Egypt  and  Russia  together  sent  9\  million 
pounds  of  eggs  to  Paris.  If  Egypt  had  sent  twice  as  many, 
the  ratio  of  this  number  to  those  sent  by  Russia  would  have 
been  1  :  18.     How  many  pounds  were  sent  by  each  country  ? 

16.  The  sum  of  the  three  dimensions  of  a  block  of  ice  is  77 
inches,  and  the  width,  22  inches,  is  a  mean  proportional  between 
the  other  two  dimensions.     Find  the  length  and  thickness. 

17.  The  ratio  of  the  length  of  a  gold  nugget  to  its  width  was 
11 :  6,  but  if  its  length  had  been  ^  of  an  inch  more,  the  ratio 
would  have  been  2  :  1.     Find  its  length  and  width. 


294 


RATIO  AND  PROPORTION 


18.  The  area  of  the  right  triangle  shown  in  Fig.  1  may  be 
expressed  either  as  |-  ah  or  as  \  ch.  Form  a  proportion  whose 
terms  shall  be  a,  b,  c,  h. 


Fig.  1. 


Fig.  2. 


Fig.  3. 


19.  In  Fig.  2,  the  perpendicular  p,  which  is  20  feet  long,  is 
a  mean  proportional  between  a  and  b,  the  parts  of  the  diame- 
ter, which  is  50  feet  long.     Find  the  length  of  each  part. 

20.  In  Fig.  3,  the  tangent  t  is  a  mean  proportional  between 
the  whole  secant  c-j-  e,  and  its  external  part  e.  Find  the 
length  of  ^,  if  e  =  9f  and  c  =  50f . 

21.  The  strings  of  a  musical  instrument  produce  sound  by 
vibrating.  The  relation  between  the  number  of  vibrations 
-Yand  N'  of  two  strings,  different  only  in  their  lengths  I  and  V, 
is  expressed  by  the  proportion 

J^i  N'  =  V:l. 

A  c  string  and  a  g  string,  exactly  alike  except  in  length, 
vibrate  256  and  384  times  per  second,  respectively.  If  the  c 
string  is  42  inches  long,  find  the  length  of  the  g  string. 

22.  If  L  and  I  are  the  lengths  of  two  pendulums  and  Tand  t 
the  times  they  take  for  an  oscillation,  then 

A  pendulum  that  makes  one  oscillation  per  second  is  approxi- 
mately 39.1  inches  long.  How  often  does  a  pendulum  156.4 
inches  long  oscillate  ? 

23.  Using  the  proportion  of  exercise  22,  find  how  many  feet 
long  a  pendulum  would  have  to  be  to  oscillate  once  a  minute. 


GENERAL   REVIEW 


404.   1.   Distinguish,  between  known  and  unknown  numbers. 

2.  When  x,  -t-,  or  both  occur  in  connection  with  -f,  — ,  or 
both  in  an  expression,  what  is  the  sequence  of  operations  ? 

Illustrate  by  finding  the  value  of:  7  —  3x24-6-7-2. 

3.  Name  and  illustrate  three  ways  of  indicating  multiplica- 
tion; two  ways  of  indicating  division. 

4r  When  is  ic"  —  ?/"  exactly  divisible  hy  x-{-y?  hj  x  —y?  ^ 

5.  When  is  a  trinomial  a  perfect  square  ?     When  is  a  frac- 
tion in  its  lowest  terms  ?     What  are  similar  fractions  ? 

6.  By  what  principle  may  cancellation  be  used  in  reducing   ^ 
fractions  to  lowest  terms  ? 

7.  Factor  the  following  by  three  different  methods  : 

(a2_2)2-a2. 

8.  Define   power;   root;  like  terms;  transposition;  simul- 
taneous equations ;  surd. 

9.  Express  the  following  without  parentheses : 

10.  What  is  the  sign  of  any  power  of  a  positive  number  ?  of 
any  even  power  of  a  negative  number  ?  of  any  odd  power  of  a 
negative  number  ? 

11.  How  may  the  involution  of  a  trinomial  be  performed  by 
the  use  of  the  binomial  formula  ? 

Illustrate  by  raising  x  -{-2y—  zto  the  third  power. 

296 


296  GENERAL   REVIEW 

12.  Explain  the  meaning  of  a  negative  integral  exponent ; 
of  a  positive  fractional  exponent ;  of  a  zero  exponent. 

13.  Define  evolution ;  binomial  surd ;  similar  surds ;  conju- 
gate surds ;  symmetrical  equation. 

14.  Is  -^2  4- V4  a  surd  ?     State  reasons  for  your  answer. 

15.  Represent  VlO  inches  by  a  line. 

16.  Why  is  it  specially  important  to  test  the  values  of  the 
unknown  number  found  in  the  solution  of  radical  equations  ? 

17.  Define  coordinate  axes;  imaginary  number;  axiom; 
coefficient;  elimination. 

18.  How  is  the  degree  of  an  equation  determined  ?  What 
is  the  degree  of  a;  +  6  =  c  ?  ofaf-hSx  =  7?  oi5x-^o(^  =  ll? 

19.  What  name  is  given  to  an  equation  of  the  first  degree  ? 
of  the  second  degree  ?  of  any  higher  degree  ? 

20.  What  is  a  pure  quadratic  equation  ?  a  complete  quad- 
ratic equation  ?     Illustrate  each. 

21.  What  is  the  root  of  an  equation  ?  What  is  the  principle 
relating  to  the  roots  of  a  pure  quadratic  equation  ? 

Illustrate  by  solving  the  following : 

7x^-5  =  28. 

22.  Give  two  methods  of  completing  the  square  in  the  solu- 
tion of  affected  quadratic  equations.  When  is  it  advantageous 
to  use  the  Hindoo  method  ? 

Solve  the  following  equation  by  each  method : 

3a^  +  5a;  =  22. 

23.  Outline  the  method  of  solving  quadratic  equations  by 
factoring. 

Illustrate  by  solving  the  following : 

2a:2__5a5  =  l2. 

24.  When  is  an  equation  in  the  quadratic  form  ?     Illustrate. 


GENERAL  REVIEW  297 

25.  What  roots  should  be  associated  when  the  roots  of  a 
system  of  equations  are  given  thus :  xz=  ±2,  y  =  =f3? 

26.  Explain  how,  in  the  solution  of  problems,  negative  roots 
of  quadratic  equations,  while  mathematically  correct,  are  often 
inadmissible. 

27.  What  is  the  advantage  of  letting  x^  =  y  in  the  graphic 
solution  of  a  quadratic  equation  of  the  form  ax^  -{•  bx  -{-  c  =  0?_ 

28.  How  does  the  graph  of  a  quadratic  equation  show  the 
fact,  if  the  roots  are  real  and  equal?  real  and  unequal? 
imaginary  ? 

29.  What  is  the  form  of  the  graph  of  a  simple  equation  ?  of 
two  inconsistent  equations  ?  of  two  indeterminate  equations  ? 

30.  What  is  the  form  of  the  graph  of  an  equation  like 
aa^  +  bx  +  c  =  0?  like  x'-^-y^^r'? 

31.  What  is  meant  by  the  minimum  point  of  a  graph  ? 

Solve  graphically  the  following  equation  and  indicate  the 
minimum  point  of  the  graph : 

32.  How  many  roots  has  a  simple  equation?  a  quadratic 
equation?  a  system  of  two  independent  simultaneous  equa- 
tions, one  simple  and  the  other  quadratic  ?  a  system  of  two 
independent  simultaneous  quadratic  equations  ? 

33.  Define  homogeneous  equation ;  antecedent ;  consequent ; 
inverse  ratio ;  proportion. 

34.  Give  and  illustrate  two  principles  relating  to  ratios. 
Upon  what  do  these  principles  depend  ? 

35.  Find  the  ratio  of  (x  —  yf  tox^  —  2xy  +  y^. 

36.  In  the  following  proportion  indicate  the  means;  the 
extremes ;  the  mean  proportional ;  the  third  proportional : 

x:y  =  y:z. 

37.  Find  a  fourth  proportional  to  3  a,  9  a,  and  5  a. 


298 


GENERAL  REVIEW 


38.  Add     X  V2/  +  y  V^  +  Va;y,    x^y^—  ^x^y  —  V^,    ■\/x^y 
■  ^xy^  —  V^,  and  y^x  —  a? V4  y  —  Vo"^. 

39.  Simplify  a-J&— a— [a-6  — (2a+6)  +  (2a— 6)  — a]-6S. 

40.  Divide  ic*  —  2/*  by  a?  —  2/  by  inspection.     Test. 

41.  Separate  a^  —  1  into  six  rational  factors. 

a^  —  6^  —  c^  —  2  6c 

42.  Reduce  — to  its  lowest  terms. 


a2_&2_|_c2_|_2ac 


43.  Simplify 

44.  Simplify 


x-\-\      \  —  x      ^  —  1 


1 


+ 


(a  —  6)(6  —  c)      (c  —  6)(c  —  a)      (c  —  a)(a  —  6) 


45.  Reduce  to  the  simplest  form :  Vl;  '\/25a'';  ^t . 

^1  +  2 

Solve  the  following  equations  for  x : 

46.  3a;2_2a?=65.  48.    V^^^  =  V^  -  1. 


47    0.4.  1_3^, 
*^-   "^+2-   2 


49.   a^  +  Va:^  +  16  =  14. 


Solve  the  following  for  the  letters  involved: 

'2a; +  3  2/ +  2;  =  9, 


-  +  -=10, 
50.     ^^      2/ 

5  +  ?=10. 
^a;     2/ 

a^-f/  =  25, 

a;4-2/  =  7. 

Solve  graphically : 
a;-2/=l, 
I  a^  4. 2/2  =  16. 


51. 


54. 


52. 


53. 


55. 


a; +  2?/ +  32;  =  13, 
I  3a; +  2/ +  2  2;  =11. 


(^ 

u^ 


+  xy  =  24, 
2/2  +  a;2/  =  12. 


2/  =  3. 


Find  the  value  of  a;  in  the  following  proportions  :^ 
56.   72:6  =  a;:4J.  67.   a; :  7.2  =  3.9  :  117. 


GENERAL  REVIEW  299 

58.  The  railways  of  the  United  States  use  annually  150 
million  tons  of  coal.  If  the  amount  used  in  drawing  trains  is 
yig-  as  much  as  goes  up  the  smokestacks,  how  much  is  used  to 
draw  trains  ? 

59.  In  one  year  about  30,000  vessels  passed  a  lighthouse  in. 
Massachusetts.  The  number  that  used  steam  was  to  the  num- 
ber of  the  remainder  as  1 : 5.     How  many  used  steam  ? 

60.  Out  of  63  bakeries  inspected  in  a  certain  city,  the  num- 
ber of  '  absolutely  clean '  ones  was  3  more  than  that  of  the 
'  fairly  clean '  ones,  and  the  number  of  '  unsanitary '  ones  was 
2  less  than  twice  that  of  the  'absolutely  clean'  ones.  Find 
the  number  of  bakeries  in  each  class. 

61.  The  number  of  parts  in  a  certain  manufacturer's  mower 
is  twice  that  in  his  horse  rake  and  ^V  that  in  his  binder.  If 
the  binder  has  3500  parts  more  than  the  rake,  how  many  parts 
has  each  machine  ? 

62.  Two  men  earned  $  3.50  one  day  for  picking  pine  needles. 
They  were  paid  25  cents  per  100  pounds.  How  many  pounds 
did  each  pick,  if  one  picked  J  as  many  as  the  other  ? 

63.  One  of  the  largest  rugs  ever  made  in  this  country  con- 
tains 3180  square  feet.  Its  length  is  7  feet  greater  than  its 
width.     What  are  its  dimensions  ? 

64.  Alfred  the  Great  measured  time  by  candles  lighted  in 
succession.  The  number  used  in  a  day  was  ^  the  number  of 
inches  in  the  length  of  each  candle,  and  each  burned  at  the 
rate  of  3  inches  per  hour.  How  many  candles  were  used  per 
day  and  how  long  was  each  ? 

65.  A  target  used  in  practice  by  the  United  States  fleet  was 
1  foot  longer  than  it  was  wide  and  18  feet  longer  than  the 
square  bull's-eye.  The  area  of  the  target  exclusive  of  the  bull's- 
eye  was  411  square  feet.     Find  the  dimensions  of  each. 

66.  A  good  operator  usually  earns  f  1.80  a  day  by  binding 
derby  hats.  If  she  bound  1  dozen  more  hats  and  received  5^ 
less  per  dozen,  she  would  earn  5  ^  less  a  day.  How  many  hats 
does  she  bind  a  day  and  how  much  does  she  receive  per  dozen  ? 


300  GENERAL   REVIEW 

67.  How  far  down  a  river  whose  current  runs  3  miles  an 
hour  can  a  steamboat  go  and  return  in  8  hours,  if  its  rate  of 
sailing  in  still  water  is  12  miles  an  hour  ? 

68.  A  person  who  can  walk  n  miles  an  hour  has  a  hours  at 
his  disposal.  How  far  may  he  ride  in  a  coach  that  travels  m 
miles  an  hour  and  return  on  foot  within  the  allotted  time  ? 

69.  The  first  copy  of  The  Sun  was  printed  on  a  sheet  5i 
inches  longer  than  it  was  wide.  If  the  length  lacked  6  inches 
of  being  twice  the  width,  find  the  dimensions  of  the  sheet. 

70.  The  Lusitania  is  26  feet  less  than  6  times  as  long  as  the 
Clermont,  and  -^^  of  the  length  of  the  Lusitania  is  11  feet 
more  than  -J-  of  the  length  of  the  Clermont.  Find  the  length 
of  each. 

71.  A  woman  has  13  square  feet  to  add  to  the  area  of  the 
rug  she  is  weaving.  She  therefore  increases  the  length  ^  and 
the  width  \j  which  makes  the  perimeter  4  feet  greater.  Find 
the  dimensions  of  the  finished  rug. 

72.  The  inventor  of  toothpicks  sold  16,250,000  during  his 
first  year  of  business.  Had  there  been  75,000  more  toothpicks 
in  each  box,  the  number  of  boxes  sold  would  have  been  15 
fewer.     How  many  boxes  did  he  sell  ? 

73.  Two  passengers  together  have  400  pounds  of  baggage 
and  are  charged,  for  the  excess  above  the  weight  allowed  free, 
40  cents  and  60  cents,  respectively.  If  the  baggage  had  be- 
longed to  one  of  them,  he  would  have  been  charged  $  1.50. 
How  much  baggage  is  one  passenger  allowed  without  charge  ? 

74.  A  railway  train,  after  traveling  2  hours  at  its  usual 
rate,  was  detained  1  hour  by  an  accident.  It  then  proceeded 
at  I  of  its  former  rate,- and  arrived  7f  hours  behind  time.  If 
the  accident  had  occurred  50  miles  farther  on,  the  train  would 
have  arrived  6^  hours  behind  time.  What  was  the  whole  dis- 
tance traveled  by  the  train  ? 


GENERAL   REVIEW  301 

405.  This  page  contains  the  questions  given  in  the  Elemen- 
tary Algebra  examination  of  the  Regents  of  the  University  of 
the  State  of  New  York  for  June,  1909. 

References  show  where  the  text  provides  instruction  neces- 
sary to  answer  these  questions. 

1.  Divide  6  aj3  +  11  a^-1  by  3  it- -14-2x2  (§§38,  108). 

2.  Find  the  prime  factors  of  1  -  ^  (§§  134, 155),  9  a*-90  a^ 

H-189a2  (§§133,142,155),  a'-^b'  (§136),  4:X*  +  Sa^y'  +  9y' 
(§154),  aa;  +  4a-4a;-16  (§  145). 

r3aj  +  8  =  4i/4-2, 

3.  Solve      i^^.^  =  3  (§§231,232). 

to       9 

Give  an  axiom  justifying  each  step  in  the  solution  (§  224). 

4.  Find  a  number  such  that  if  it  is  added  to  1,  4,  9,  16, 
respectively,  the  results  will  form  a  proportion  (§  403). 


5.  Solve  Vx -f  1  +  Va: -  2  =  V2  a; 4-  3  (§§  362,  353). 

6.  Find   the   square  root  of  ^'+^~^-f  — -  — +  1 

9        lo         o       2o       5 

(§  280). 

7.  Simplify  i/iree  of  the  following:   ■\/—125x^,   V^-^, 

■^1087"*  (§  301),   </|  X  </^  (§§  314,  315),   V75  -  4V243 -f- 
2Vi08(§§  311^13). 

^  ,        (  x^ -\- 2  xy  =  55,     (Elimination  of  similar  terms, 
l2ar^-a^  =  35.      page  270.) 

9.  If  the  speed  of  a  railway  train  should  be  lessened  4 
miles  an  hour,  the  train  would  be  half  an  hour  longer  in  going 
180  miles.     Find  the  rate  of  the  train  (§§  205,  215). 

10.  If  the  greater  of  two  numbers  is  divided  by  the  less,  the 
quotient  is  2  and  the  remainder  is  3.  The  square  of  the 
greater  number  exceeds  6  times  the  square  of  the  less  by  25. 
Find  the  numbers  (§  374). 


302  GENERAL  REVIEW 

State  of  New  York  Regents'  exai^ination  for  June,  1910 : 

1  +  -^ 

,      a-        ^'f  ^  +  1       (a-\-iy  —  a^    /nonfw 

1.  Simplify  —^       a'-l (§  ^^^)- 

1  -f-a 

2.  Find  the  prime  factors  otfour  of  the  following:  a"*—  16 
(§§  134,  149)  ;  20  a;2  -  60  ajy  +  45  y'  (§§  133,  138)  ;  6  aa;  +  10  a^ 

-2160^-35  62/ (§145);  ?/' - 2/ +  i  (§  138) ;  1-^(§136). 


3.    Solve   |«^  +  ^^  =  ^. 

\6a;  +  a2/  =  n  (§233). 


8 


4.  Reduce  eacA  of  the  following  to  its  simplest  form: 
(V3  +  5V2)(2V3  +  3V2)  [§  315];  V108-V432  (§  317); 
V|  (§  302)  ;  VT8  +  V50  -  V72  (§  313) ;  ^/2  x  ^4  (§  315). 

5.  It  required  as  many  days  for  a  number  of  men  to  dig  a 
trench  as  there  were  men ;  if  there  had  been  6  men  more  the 
work  would  have  been  done  in  8  days.  Find  the  number  of 
men  (§§  166,  354). 

6.  Solvea,'2  +  5aa;  =  14a2(§§350,  351). 

7.  Solve  {2  «^  4-^:2/ =  24, 

1     x^-f  =  5  (§368). 

8.  The  length  of  a  picture  at  the  inner  edge  of  the  frame 
is  twice  its  width ;  the  frame  is  4  inches  wide  and  has  an  area 
of  328  square  inches.  Find  the  dimensions  of  the  picture 
(§§  125  and  215  or  §  234). 

9.  Solve  V^T7  +  V^  =  7  (§ §  333,  334). 

10.  The  area  of  a  certain  square  is  f  of  the  area  of  a  certain 
other  square  and  its  side  is  J  of  a  yard  less.  Find  the  side  of 
each  square  (§  374). 

11.  Two  quarts  of  alcohol  are  mixed  with  5  quarts  of  water. 
Find  the  number  of  quarts  of  alcohol  that  must  be  added  to 
make  the  mixture  three  fourths  alcohol  (§  215). 

12.  Define  exponent,  surd,  term,  reciprocal,  elimination 
(§§  9,  298,  3,  196,  224,  respectively,  and  the  glossary). 


GENERAL   REVIEW  303 

406.  The  following  questions  were  asked  in  the  Minnesota 
High  School  Board  examination  for  Elementary  Algebra  in 
June,  1910. 

1.  Simplify  a_[_j_(-3a-2a-6)S]  (§116). 

2.  A's  age  exceeds  B's  by  20  years.  Ten  years  ago  A  was 
twice  as  old  as  B.  Find  the  age  of  each  (§§  125,  215;  also 
§  234). 

3.  Factor: 

(1)  6bx-  15  ab  -  4.  dx  + 10  ad  (§  145). 

(2)  x'-Wy'  (§§  134,149). 

(3)  1-lSx-GSx^  (%  144). 

(4)  64a«  +  8  (§§  133,136). 

(5)  3:^-27  (§  136). 
x  —  y       X—  a    ^    y—  b 


4.   Find   the   algebraic   sum   of    ^ f 


(§§191,192).  ^^        «^  +  ^^     «^  +  ^^ 

5.  Find  the  produet  of  ^^tf?  x    ,  ^^^    ^  (§§  193, 194). 

a-*  — 16     a^  —  3  «  +  9 

6.  Find  the  quotient  of  ^-=-^  -  (a  -  6)    (§§  172,  195-198). 

7.  Seven  men  and  5  boys  earn  ^11.25  per  day  and  at  the 
same  wages  4  men  and  12  boys  earn  $  11  per  day.  Find  the 
wages  of  each  per  day  (§§  104,  234). 


8.  Simplify  V16a  +  V81  a  +  Vl44 a^^^  (§  313). 

9.  Multiply  4V8- V32-f2V50  by  V2  (§  315). 

10.  The  distance  from  Chicago  to  Minneapolis  is  420  miles. 
By  increasing  the  speed  of  a  certain  train  7  miles  per  hour, 
the  running  time  is  decreased  2  hours.  Find  the  speed  of  the 
train  (§  354). 

11.  Find  two  numbers  in  the  proportion  of  3  to  5  whose 
sum  is  160  (§  403). 

2m-3+i 

12.  Simplify       ^^_^'''   (§  200). 

m 


304  GENERAL  REVIEW 

Minnesota  High  School  Board  examination  for  May,  1907 : 

1.  Define  any  five:  Algebra,  term,  root,  power,  binomial, 
reciprocal  (glossary). 

2.  From  the  sum  of  a^ -\- 2 a^b -^ 5 ah^ -\- S  b^  said  2a^-Sa^b 
-f  3  a52  _  53^  ta]^g  their  difference  (§§  62,  63,  66), 

3.  Express  in  simplest  form  : 
-(4:X-7y-^4:)-\-2x-(Sy-4:y-x  +  4:)\  (§  116). 

4.  Divide  3  ic^n  _|.  ^3  ^n-i  _^  j^5  ^2n-2  ^  9  ^n-3  ^y  x'^-^-Sx^-^ 
(§  108). 

5.  Factor   any   five:     (a)  a^^-lOaf +  16    (§§    142,   159); 
(b)  ici2_y2  (§§  134^  ;^3g^  ;^49>^.  ^^>^  l2a^  +  2a«/-2/  (§§  133, 

144);   (d)  x3+15a^-a;-15(§§  145,134);    (e)  a'-16  +  b'+2ab 

(§151);  (/)  0:^  +  4^^  (§§153,  154). 

6.  Expand  by  the  binomial  formula,  showing  all  the  steps ; 

(2a-a;)5[§§  265,266]. 

7.  Solve  for  » : 


2^±J _  ^-zl  =  7y-4a;  +  36  ^^ ^  23I,  232). 
2  8  16 

8.  One  half  of  A's  money  is  equal  to  B's,  and  five  eighths 
of  B's  is  equal  to  C's;  together  they  have  $1450.  How  much 
has  each?  (§§  47,  75,  125,  205,  215). 


9.   Simplify  \  "^     ^t""  (§  200). 


1  — a     l-f-ic 

10.  A  man  bought  a  suit  of  clothes  for  $24  and  paid  for  it 
in  two-dollar  bills  and  fifty-cent  pieces,  giving  twice  as  many 
coins  as  bills.    How  many  bills  did  he  give  ?  (§§  125,  215,  234). 

11.  Five  years  ago  the  sum  of  the  ages  of  A  and  B  was  40 
years.  B  is  now  four  times  as  old  as  A.  What  is  the  present 
age  of  each  ?  (§§  125,  215;  also  §  234). 


FACTORS   AND   MULTIPLES 


407.  This  chapter  gives  a  brief  treatment  of  highest  common 
factor  (§  183)  and  lowest  common  multiple  (§  189)  for  the 
benefit  of  any  who  may  desire  a  little  more  work  in  these 
topics  than  their  application  affords  in  fractions,  the  only 
place  in  elementary  algebra  where  they  are  applied. 

HIGHEST   COMMON   FACTOR 

408.  An  expression  that  is  a  factor  of  each  of  two  or  more 
expressions  is  called  a  common  factor  of  them. 

409.  The  common  factor  of  two  or  more  expressions  that 
has  the  largest  numerical  coefficient  and  is  of  the  highest 
degree  is  called  their  highest  common  factor. 

The  common  factors  of  4  a^h^  and  6  cfih  are  2,  a,  6,  a^,  2  a,  2  6,  2  a^, 
ah,  2  dby  a%,  and  2  a-h  with  sign  +  or  - .  Of  these,  2  a%  (or  -  2  a%) 
has  the  largest  numerical  coefficient  and  is  of  the  highest  degree,  and  is 
therefore  the  highest  common  factor. 

The  highest  common  factor  may  be  positive  or  negative,  but  usually 
only  the  positive  sign  is  taken. 

The  highest  common  factor  of  two  or  more  expressions  is 
equal  to  the  product  of  all  their  common  prime  factors. 

410.  Expressions  that  have  no  common  prime  factor,  except 
1,  are  said  to  be  prime  to  each  other. 

EXERCISES 

411.  1.  Find  the  h.  c.  f.  of  12  a^hh  and  32  a^ftV. 

Solution 
The  arithmetical  greatest  common  divisor  or  highest  common  factor  of 
12  and  32  is  4.    The  highest  common  factor  of  a^V^c  and  of  a^h^ifi  is  a^h'^c. 
Hence,  h.  c.  f.  =  4  a%'^c. 

MILNE'S  IST  YR.  ALG.  —  20        305 


306  FACTORS   AND  MULTIPLES 

KuLE.  —  To  the  greatest  common  divisor  of  the  numerical  co- 
efficients annex  each  common  literal  factor  with  the  least  exponent 
it  has  in  any  of  the  expressions. 

Find  the  highest  common  factor  of : 

2.  10  a^f,  10  x'f,  and  15  xy*z. 

3.  70a%',21a*b\a,ndS5a'b'. 

4.  8  mhi%  28  my,  and  56  m'nK 

5.  4:b^cd,  6  6V,  and  24  a6c^ 

6.  3(a  +  6)2  and  6(a  +  6)3. 

7.  6(a  +  6)2and4(a-}-6)(a-6). 

8.  12  (a  -  xf,  6  (a  -  x)'',  and  (a  -  x)\ 

9.  10 (x-yy^ 'dud  15 (z-y)(x-yf. 

10.    Whatis  theh.c.f.of3a^-3a;2/2and6a:3_i2a^2/  +  6a;y29 

PROCESS 

Sa^-Sxy^  =Sx(x  +  y)(x-y) 

6  x" -  12 xry -\- 6  xy^  =  2- 3 x(x-y)(x-y) 
.-.  h.c.f.  =  3a;(a;  —  y) 

Explanation.  —  For  convenience  in  selecting  the  common  factors,  the 
expressions  are  resolved  into  their  simplest  factors. 

Since  the  only  common  prime  factors  are  3,  x,  and  (x  —  y),  the  highest 
common  factor  sought  (§  409)  is  their  product  Sx{x  —  y). 

Find  the  highest  common  factor  of : 

11.  a^  —  x^  3ind  a^  —  2  ax -{- x^. 

12.  a"^ -b^  and  a^ +  2  ab-\-b^' 

13.  x^-\-y^  and  a^ -\- 2  xy  -{- y^. 

14.  a^-2x-15  2iudx^-x-20. 

15.  a2  +  7a  +  12anda2H-5aH-6.     ' 


FACTORS  AND  MULTIPLES  307 

Find  the  highest  common  factor  of : 

16.  aj^  +  a^^2  +  2/*  and  a^  +  icy  4- 2/^ 

17.  ic^  +  2/■^  ^  +  ^>  and  x^y  +  xy"^. 

18.  a*  +  tt26*  +  68  and  3a2-3a62  +  36*. 

19.  ax  —  y -\-xy—a2LYi(i  ay? -{-x^y —  a  —  y. 

20.  a^b  —  b  —  ah  +  c  and  ab  —  ac  —  b-\-c. 

21.  l-4a^,  l-f-2a;,  and4a-16aa^. 

22.  24a^/  +  8a;y  and  8a:3y3_3aj22^^ 

23.  6ar^  +  a;-2  and  2a^-lla--f-5. 

24.  17  aftc'^d^  -  51  a^ftc^d^  and  aftc^d*^  -  3  a^b(^d. 

25.  38  xyz  -  95  a^y^^  ^nd  34  xy^z  -  85  ic^y^s 

26.  6r^  +  10/s-4  j-^s^  and  2 r'  +  2  ?^s - 4 ? V. 

27.  a;*-ar^-2ar,  a;*  -  2  a:^  -  3  a^,  and  a;*  -  3  «»  -  4  a:^. 

28.  7  Z3«3  +  35Z-'^  +  42  /^  and  7  Z^^  -f-  21 Z^^-  28  J^f-UW, 

29.  a^-f-a2 -524.2  aa;,  a;2-a2+62  4_2  6a;,  and  a?-a^-b''-2ab. 

30.  a^-6a;  +  5anda;«-5a^  +  7a;-3. 

Suggestion.  —  Apply  the  factor  theorem  to  the  second  expression. 

31    a^-4a;H-3anda^  +  ar'-37a;H-35. 

32.  Q-n^andn^-n-e. 

Suggestion.  —  Change  9  —  n^  to  —  (n"  —  9)  =  -  (n  +  3)  (n  —  3). 

33.  l-a^anda^-6a,-2-9a;  +  14. 

34.  (9-a^)2anda^H-2ar'-9x-18. 
Suggestion.     (9  -  x^)^  -  (x^  —  9)2. 

35.  (4-c2)2andc3  +  9c2  +  26c-i-24. 

36.  xy  -  y^,  -  (f  -  x^y),  and  x^y  -  xy\ 

37.  16-s^2s-s^  ands2-4s  +  4. 

38.  2/*  —  a;^  a,-^  4- y*,  and  2/2  +  2  2/a;  +  a;^ 

39.  {y-xf{n-mf  and  (a^^i/ - 2/^) {m-n -2mn'  +  n^) . 


308  FACTORS   AND  MULTIPLES 

LOWEST  COMMON  MULTIPLE 

412.  An  expression  that  exactly  contains  each  of  two  or 
more  given  expressions  is  called  a  common  multiple  of  them. 

6  ahx  is  a  common  multiple  of  a,  3  &,  2  x,  and  6  ahx.  These  numbers 
may  have  other  common  multiples,  as  12  ahx,  6  a^h'^x,  18  a^fex^,  etc. 

413.  The  expression  having  the  smallest  numerical  coef- 
ficient and  of  loivest  degree  that  will  exactly  contain  each  of 
two  or  more  given  expressions  is  called  their  lowest  common 
multiple. 

6  abx  is  the  lowest  common  multiple  of  a,  3  6,  2  x,  and  6  abx. 
The  lowest  common  multiple  may  have  either  sign  +  or  — ,  though 
usually  only  the  positive  sign  is  taken. 

The  lowest  common  multiple  of  two  or  more  expressions  is 
equal  to  the  product  of  all  their  different  prime  factors,  each 
factor  being  used  the  greatest  number  of  times  it  occurs  in  any 
of  the  expressions. 

EXERCISES 

414.  1.  What  is  the  1.  c.  m.  of  12  x^y^,  6  a^xy^y  and  8  axyz^  ? 

Solution 

The  lowest  common  multiple  of  the  numerical  coefficients  is  found  as 
in  arithmetic.     It  is  24. 

The  literal  factors  of  the  lowest  common  multiple  are  each  letter  with 
the  highest  exponent  it  has  in  any  of  the  given  expressions.  They  are, 
therefore,  a^,  x^,  y'^,  and  z^. 

The  product  of  the  numerical  and  literal  factors,  24:  a^x^y^z^,  is  the 
lowest  common  multiple  of  the  given  expressions. 

Find  the  lowest  common  multiple  of : 

2.  a^oi^yj  a^xy^,  and  aa^y. 

3.  10  a^ftV,  5  ab\  and  25  b^c^d\ 

4.  16  a'b%  24  (^de,  and  36  a*b'd^e^ 

5.  18  a'br',  12 p^r,  and  54  a^Vg. 


FACTORS   AND  MULTIPLES  309 

6.   What  is  the  1.  c.  m.  oi  x^  —  2xy-\-  y^,  y^  —  a^,  and  a^  +  y^? 

PROCESS 

a:^-2xy-{-y^  ={x--y){x-y) 

y^ -  x^  =  -  (xr^ -y^)  =  -  (x-\-y){x-y) 

.-.  1.  c.  III.  =  {x-  y)\x  ■\-y){^-  xy  +  y^) 
=  (x-yy(a^  +  f) 

EuLE.  —  Factor  the  expressions  into  their  prime  factors. 

Find  the  product  of  all  their  different  prime  factors,  using  each 
factor  the  greatest  number  of  times  it  occurs  in  any  of  the  given 
expressions. 

The  factors  of  the  1.  c.  m.  may  often  be  selected  without  separating  the 
expressions  into  their  prime  factors. 

Find  the  lowest  common  multiple  of  : 

7.  Q^  —  y^  and  o^  -\-2xy -\-y'^. 

8.  X?  —  y^  and  x^  —  2  xy  -{-  y^. 

9.  x^  —  y^,  x^  -\-  2  xy  -\-  y^,  and  x^  —  2xy-\-y^. 

10.  a'-n^2iudLSa^-^6ahi+San'. 

11.  ic^-1  anda2i»2  +  a'-6V-62. 

12.  a^  -\-l,  ab  —  b,  a^  +  a,  and  1  —  a^. 

13.  2  X -\- y,  2  xy  —  y^,  and  4:  x^  —  y^. 

14.  1 -\- X,  X  —  x^,  1 -j- x^,  and  a^(l  —  a?). 

15.  2a;  +  2,  5a;-5,3a;-3,  and  0.-2-1. 

16.  1662_1,  1262^35^205-5,  and  26. 

17.  l-2x'-\-x*,(l-xy,Rndl  +  2x  +  x'. 

18.  b''-5b-{-6,  b^^7b  +  10,  and  6^ -  10 6  + 16. 


310  FACTORS   AND  MULTIPLES 

Find  the  lowest  common  multiple  of : 

19.  oc^ -\- 7  X  —  S,  a^  —  1,  X -{- x^,  smd  S  aa^  —  6  ax -\- 3  a. 

20.  x^  —  a%a-2  x,  a^  +  2  ax,  and  a^ -  3  a^ic  -f  2  ao?. 

21.  m^  —  ar^,  m^  +  ma;,  7/i^  -|-  mx  -f-  a^,  and  (m  +  a;)  x^. 

22.  2-3a;  +  a^,  a^  +  4a;  +  4,  a^  +  3a;  +  2,  and  1  -  a^. 

23.  a^  —  2/^  ic"*  +  ^y^  -\-y^,  ^  +  y^,  and  a^  +  a;?/  +  y^. 

24.  a^  +  a;^2/  +  ^V^  +  2/^  ^^^  x^  —  a^2/  +  ^2/^  ~  2/^- 

25.  a^  _|_  4  o[  _^  4^  0^2  —  4,  4  -  a^,  and  a^  — 16. 

26.  a^  -  (6  +  c)2,  62  -  (c  +  af,  and  c^  -  (a  +  6)2. 

27.  2  (aa^  -  ar^)2,  3  x {oj'x  -  ar^)^  and  6  (aV  -  a^) . 

28.  {yz^  —  xyzf,  y^  {xz^  —  x^,  and  a?z^  +  2  a;2^  +  z^. 

Suggestion.  —  In  solving  the  following,  use  the  factor  theorem. 

29.  a^-6a.-2  +  llaj-6  anda;^-9a;2  +  26a;-24. 

30.  a^-5a;2_4^_^20anda^4-2a;2-25a;-50. 

31.  a^-4a^  +  5a;-2anda^-8a^  +  21a;-18. 

32.  a^  +  5a^4-7a;  +  3  and  a^  —  7a^  —  5a;  +  75. 

33.  a^4-2a^-4a;-8,  a^-a^ -8a; +  12,  a^  +  4a;2- 3  a;- 18. 


GLOSSARY 


Abscissa.     A  distance  measured  along  or  parallel  to  the  x-axls. 

Absolute  Term.     A  term  that  does  not  contain  an  unknown  number. 

Absolute  Value.     The  value  of  a  number  without  regard  to  its  sign. 

Addends.    Numbers  to  be  added. 

Addition.  The  process  of  finding  a  simple  expression  for  the  algebraic 
sum  of  two  or  more  numbers. 

Affected  Quadratic.    A  quadratic  equation  that  contains  both  the  second 

and  first  powers  of  one  unknown  number. 

Algebra.  That  branch  of  mathematics  which  treats  of  general  numbers 
and  the  nature  and  use  of  equations.  It  is  an  extension  of  arithmetic  and 
it  uses  both  figures  and  letters  to  express  numbers. 

Algebraic  Expression.     A  number  represented  by  algebraic  symbols. 

Algebraic  Numbers.  Positive  and  negative  numbers,  whether  integers 
or  fractions. 

Algebraic  Sum.    The  result  of  adding  two  or  more  algebraic  numbers. 

Antecedent.     The  first  term  of  a  ratio. 

Arrangement.  When  a  polynomial  is  arranged  so  that  in  passing  from 
left  to  right  the  several  powers  of  some  letter  are  successively  higher  or 
loioer^  the  polynomial  is  said  to  be  arranged  according  to  the  ascending 
or  descending  powers,  respectively,  of  that  letter. 

Axes  of  Reference.  Two  straight  lines  that  intersect,  usually  at  right 
angles,  used  to  locate  a  point  or  points  in  a  plane. 

Axiom.     A  principle  so  simple  as  to  be  self-evident. 

Binomial.     An  algebraic  expression  of  two  terms. 

^    Binomial  Formula.     The  formula  or  principle  by  means  of  which  any 
indicated  power  of  a  binomial  may  be  expanded. 

Binomial  Quadratic  Surd.  A  binomial  surd  whose  surd  or  surds  are  of 
the  second  order. 

311 


312  GLOSSARY 

Binomial  Surd.     A  binomial,  one  or  both  of  whose  terms  are  surds. 

Clearing  an  Equation  of  Fractions.  The  process  of  changing  an  equa- 
tion containing  fractions  to  an  equation  without  fractions. 

Coefficient.  When  one  of  the  two  factors  into  which  a  number  can  be 
resolved  is  a  known  number,  it  is  usually  written  first  and  called  the 
coefficient  of  the  other  factor. 

In  a  broader  sense,  either  one  of  the  two  factors  into  which  a  number 
can  be  resolved  may  be  considered  the  coefficient  of  the  other. 

Co-factor.     Same  as  Coefficient. 

Common  Factor.     A  factor  of  each  of  two  or  more  numbers. 

Common  Multiple.  An  expression  that  exactly  contains  each  of  two 
or  more  given  expressions. 

Complete  Quadratic.     Same  as  Affected  Quadratic. 

Complex  Fraction.  A  fraction  one  or  both  of  whose  terms  contains  a 
fraction. 

Compound  Expression.     Same  as  Polynomial. 

Conditional  Equation.  An  equation  that  is  true  for  only  certain  values 
of  its  letters. 

Conjugate  Surds.  Two  binomial  quadratic  surds  that  differ  only  in  the 
sign  of  one  of  the  terms. 

Consequent.    The  second  term  of  a  ratio. 

Consistent  Equations.     Same  as  Simultaneous  Equations. 

Coordinates.     See  Bectangular  Coordinates. 

Couplet.     The  two  terms  of  a  ratio. 

Cube.     Same  as  Third  Power. 

Cube  Root.    One  of  the  three  equal  factors  of  a  number. 

Cubic  Surd.    A  surd  of  the  third  order. 

Degree  of  an  Expression.  The  term  of  highest  degree  in  any  rational 
integral  expression  determines  the  degree  of  the  expression. 

Degree  of  a  Term.  The  sum  of  the  exponents  of  the  literal  factors  of  a 
rational  integral  term  determines  the  degree  of  the  term. 

Denominator.    The  divisor  in  an  algebraic  fraction. 


GLOSSARY  313 

Dependent  Equations.  Two  or  more  equations  that  express  the  same 
relation  between  the  unknown  numbers  involved  are  often  called  depend- 
ent equations,  for  each  may  be  derived  from  any  one  of  the  others. 

Derived  Equations.    Same  as  Dependent  Equations. 

Difference.     The  result  of  subtracting  one  number  from  another. 
That  is,  the  difference  is  the  algebraic  number  that  added  to  the  subtra- 
hend gives  the  minuend. 

Dissimilar  Fractions.     Fractions  that  have  different  denominators. 

Dissimilar  Terms.  Terms  that  contain  different  letters  or  the  same 
letters  with  different  exponents. 

Dividend.     In  division,  the  number  that  is  divided. 

Division.     The  process  of  finding  one  of  two  factors  when  their  product 

and  one  of  the  factors  is  given. 

Divisor.     In  division,  the  number  by  which  the  dividend  is  divided. 

Elimination.  The  process  of  deriving  from  a  system  of  simultaneous 
equations  another  system  involving  fewer  unknown  numbers. 

Entire  Surd.     A  surd  that  has  no  rational  coefficient  except  unity. 

Equation.     A  statement  of  the  equality  of  two  numbers  or  expressions. 

Equation  of  the  First  Degree.     Same  as  Simple  Equation. 

Equation  of  the  Second  Degree.    Same  as  Quadratic  Equation. 

Equivalent  Equations.     Two  equations  that  have  the  same  roots,  each 

equation  having  all  the  roots  of  the  other. 

Even  Root.     A  root  whose  index  is  an  even  number. 

Evolution.     The  process  of  finding  any  required  root  of  a  number. 

Exponent.  A  small  figure  or  letter  placed  at  the  right  and  a  little 
above  a  number  to  indicate  how  many  times  the  number  is  to  be  used  as 
a  factor. 

Extremes.     The  first  and  fourth  terms  of  a  proportion. 

Factor.   Each  of  two  or  more  numbers  whose  product  is  a  given  number. 

Factoring.     The  process  of  separating  a  number  into  its  factors. 

Formula.    An  expression  of  a  principle  or  a  rule  in  symbols. 


314  GLOSSARY 

Fourth  Proportional.  The  fourth  number  of  four  different  numbers 
that  form  a  proportion. 

Fourth  Root.     One  of  the  four  equal  factors  of  a  number. 

Fraction.   In  algebra,  an  indicated  division. 

Fractional  Equation.  An  equation  that  involves  an  unknown  number 
in  any  denominator. 

Fractional  Expression.   An  expression,  any  term  of  which  is  a  fraction. 

Fulcrum.     The  point  or  edge  upon  which  a  lever  rests. 

General  Number.  A  literal  number  to  which  any  value  may  be 
assigned. 

Graph.  A  picture  (line  or  lines)  every  point  of  which  exhibits  a  pair 
of  corresponding  values  of  two  related  quantities. 

Graph  of  an  Equation.  The  line  or  lines  containing  all  the  points,  and 
only  those,  whose  coordinates  satisfy  a  given  equation. 

Higher  Equation.  An  equation  that  contains  a  higher  power  of  the 
unknown  number  than  the  second. 

Highest  Common  Factor.  The  common  factor  of  two  or  more  expres- 
sions that  has  the  largest  numerical  coefficient  and  is  of  the  highest 
degree. 

It  is  equal  to  the  product  of  all  the  common  factors  of  the  expressions. 

Homogeneous  Equation.  An  equation  all  of  whose  terms  are  of  the 
same  degree  with  respect  to  the  unknown  numbers. 

Identical  Equation.  An  equation  whose  members  are  identical,  or 
such  that  they  may  be  reduced  to  the  same  form. 

Identity.     Same  as  Identical  Equation. 

Imaginary  Number.  A  number  that  involves  an  indicated  even  root  of 
a  negative  number. 

Incomplete  Quadratic.    Same  as  Pure  Quadratic. 

Inconsistent  Equations.  Two  or  more  equations  that  are  not  satisfied 
in  common  by  any  set  of  values  of  the  unknown  numbers. 

Independent  Equations.  Two  or  more  equations  that  express  different 
relations  between  the  unknown  numbers  involved,  and  so  cannot  be  re- 
duced to  the  same  equation. 

Indeterminate  Equation.  An  equation  that  is  satisfied  by  an  unlimited 
number  of  sets  of  values  of  its  unknown  numbers. 


GLOSSARY  315 

Index  of  a  Power.     Same  as  Exponent. 

Index  of  a  Root.  A  small  figure  or  letter  written  in  the  opening  of  a 
radical  sign  to  indicate  what  root  of  a  number  is  sought. 

Integer.     Same  as  miole  Number. 

Integral  Equation.  An  equation  that  does  not  involve  an  unknown 
number  in  any  denominator. 

Integral  Expression.    An  expression  that  contains  no  fraction. 

Inverse  Ratio.     Same  as  Beciprocal  Ttatio. 

Involution.  The  process  of  finding  any  required  power  of  an  expres- 
sion. 

Irrational  Equation.  An  equation  involving  an  irrational  root  of  an 
unknown  number. 

Irrational  Expression.  An  expression  that  contains  an  irrational 
number. 

Irrational  Number.  A  number  that  cannot  be  expressed  as  an  integer 
or  as  a  fraction  with  integral  tenns. 

Known  Number.     A  general  number  or  a  number  whose  value  is  known. 

Lever.    Any  sort  of  a  bar  resting  on  a  fixed  point  or  edge. 

Like  Terms.    Same  as  Similar  Terms. 

Linear  Equation.     Same  as  Simple  Equation. 

Literal  Coefficient.     A  coefficient  composed  of  letters. 

Literal  Equation.  An  equation  one  or  more  of  whose  known  numbers 
is  expressed  by  letters. 

Literal  Numbers.     Letters  that  are  used  for  numbers. 

Lowest  Common  Denominator.  The  denominator  of  lowest  degree, 
having  the  least  numerical  coefficient,  to  which  two  or  more  fractions  can 
be  reduced. 

It  is  equal  to  the  lowest  common  multiple  of  the  given  denominators. 

Lowest  Common  Multiple.  The  expression  having  the  smallest  nu- 
merical coefficient  and  of  lowest  degree  that  will  exactly  contain  each  of 
two  or  more  given  expressions. 

Lowest  Terms.  When  the  terms  of  a  fraction  have  no  common  factor, 
the  fraction  is  said  to  be  in  its  lowest  terms. 


316  GLOSSARY 

Mean  Proportional.  A  number  that  serves  as  both  means  of  a  propor- 
tion. 

Means.    The  second  and  third  terms  of  a  proportion. 

Members  of  an  Equation.  In  an  equation,  the  number  on  the  left  of 
the  sign  of  equality  is  called  the  first  member  of  the  equation,  and  the 
number  on  the  right  is  called  the  second  member. 

Minimum  Point  of  a  graph.  The  point  of  a  graph  that  has  the  alge- 
braically least  ordinate. 

Minuend.  In  subtraction,  the  number  from  which  the  subtraction  is 
made. 

Mixed  Coefficient.    A  coefficient  composed  of  both  figures  and  letters. 

Mixed  Expression.  An  expression  some  of  whose  terms  are  integral 
and  some  fractional. 

Mixed  Number.     Same  as  Mixed  Expression. 

Mixed  Surd.    A  surd  that  has  a  rational  coefficient. 

Monomial.     An  algebraic  expression  of  one  term  only. 

Multiplicand.     In  multiplication,  the  number  multiplied. 

Multiplication.  When  the  multiplier  is  a  positive  integer,  the  process 
of  taking  the  multiplicand  as  many  times  as  there  are  units  in  the  mul- 
tiplier. 

In  general,  the  process  of  finding  a  number  that  is  obtained  from  the 
multiplicand  just  as  the  multiplier  is  obtained  from  unity. 

Multiplier.  In  multiplication,  the  number  by  which  the  multiplicand 
is  multiplied. 

Negative  Number.     A  number  less  than  zero. 

Negative  Term.    A  term  preceded  by  — . 

Numerator.    The  dividend  in  an  algebraic  fraction. 

Numerical  Coefficient.     A  coefficient  composed  of  figures. 

Numerical  Equation.  An  equation  all  of  whose  known  numbers  are 
expressed  by  figures. 

Odd  Root.    A  root  whose  index  is  odd. 

Order  of  a  radical  or  of  a  surd  is  indicated  by  the  index  of  the  root  or 
by  the  denominator  of  the  fractional  exponent. 

Ordinate.    A  distance  measured  along  or  parallel  to  the  y-axis. 


GLOSSARY  317 

Origin.     The  intersection  of  the  axes  of  reference. 

Perfect  Square.  An  expression  that  may  be  separated  into  two  equal 
factors. 

Polynomial.    An  algebraic  expression  of  more  than  one  term. 

Positive  Number.     A  number  greater  than  zero. 

Positive  Term.    A  term  preceded  by  +,  expressed  or  understood. 

Power  of  a  Number.  The  product  obtained  when  the  number  is  used 
a  certain  number  of  times  as  a  factor. 

Prime  Number.     A  number  that  has  no  factors  except  itself  and  1. 

Prime  to  Each  Other.  Expressions  that  have  no  common  prime  factor 
except  1  are  said  to  be  prime  to  each  other. 

Principal  Root.  A  real  root  of  a  number  that  has  the  same  sign  a.s  the 
number  itself. 

Product.     The  result  of  multiplying  one  number  by  another. 

Proportion.     An  equality  of  ratios. 

Pure  Quadratic.  An  equation  that  contains  only  the  second  power  of 
the  unknown  number. 

Quadratic  Equation.  An  equation  that,  when  simplified,  contains  the 
square  of  the  unknown  number,  but  no  higher  power. 

Quadratic  Form.  An  expression  that  contains  but  two  powers  of  an 
unknown  number  or  expression,  the  exponent  of  one  power  being  twice 
that  of  the  other. 

Quadratic  Surd.     A  surd  of  the  second  order. 

Quotient.    The  result  of  dividing  one  number  by  another. 

Radical.     An  indicated  root  of  a  number. 

Radical  Equation.     Same  as  Irrational  Equation. 

Radical  Sign.     Same  as  Boot  Sign. 

Radicand.     A  number  whose  root  is  required. 

Ratio.  The  relation  of  two  numbers  that  is  expressed  by  the  quotient 
of  the  first  divided  by  the  second. 

Rational  Expression.  An  expression  that  contains  no  irrational 
number. 

Rationalization.  The  process  of  multiplying  an  expression  containing 
a  surd  by  any  number  that  will  make  the  product  rational. 


318  GLOSSARY 

Rationalizing  Factor.  The  factor  by  which  a  surd  expression  is  multi- 
plied to  render  the  product  rational. 

Rationalizing  the  Denominator.  The  process  of  reducing  a  fraction 
having  an  irrational  denominator  to  an  equal'  fraction  having  a  rational 
denominator. 

Rational  Number.  A  number  that  is,  or  may  be,  expressed  as  an 
integer  or  as  a  fraction  with  integral  terms. 

Real  Number.  A  number  that  does  not  involve  the  even  root  of  a 
negative  number. 

Reciprocal  of  a  number  is  1  divided  by  the  number. 

Reciprocal  of  a  Fraction  is  the  fraction  inverted  or  1  divided  by  the 
fraction. 

Reciprocal  Ratio.  The  ratio  of  the  reciprocals  of  two  numbers  is  called 
the  reciprocal  ratio  of  the  numbers. 

Rectangular  Coordinates.  The  abscissa  and  ordinate  of  a  point  referred 
to  two  perpendicular  axes  are  called  the  rectangular  coordinates  of  the 
point. 

Reduction.  The  process  of  changing  the  form  of  an  expression  with- 
out changing  its  value. 

Remainder  in  subtraction.     Same  as  Difference. 

Root  of  an  Equation.     Any  number  that  satisfies  the  equation. 

Root  of  a  Number.  When  the  factors  of  a  number  are  all  equal,  one  of 
the  factors  is  called  a  root  of  the  number. 

Root  Sign.  The  symbol  y/  written  before  a  number  denotes  that  a 
root  of  the  number  is  sought. 

Satisfied.  When  an  equation  is  reduced  to  an  identity  by  the  substi- 
tution of  certain  known  numbers  for  the  unknown  numbers,  the  equation 
is  said  to  be  satisfied. 

Second  Power.  When  a  number  is  used  twice  as  a  factor,  the  product 
is  called  the  second  power  of  the  number. 

Second  Root.     Same  as  Square  Boot 

Sign  of  Addition  is  +,  read  ^  plus.'' 

Sign  of  a  Fraction.  The  sign  written  before  the  dividing  line  of  a 
fraction. 


GLOSSARY  319 

Sign  of  Continuation  is  •••,  read  '  and  so  on'  ov  ^  and  so  on  to.' 
Sign  of  Deduction  Ls  .  •.,  read  '  therefore '  or  '  hence.' 

Sign  of  Division  is  -^,  read  '  divided  by.' 

Division  is  also  indicated  by  a  fraction,  the  numerator  being  the 
dividend  and  the  denominator  the  divisor. 

Sign  of  Equality  is  =,  read  '  is  equal  to'  or  <■  equals.'' 

Sign  of  Multiplication  is  x  or  the  dot  (•),  read  'multiplied  by.'' 

Multiplication  is  also  indicated  by  the  absence  of  sign. 

Sign  of  Ratio  is  a  colon  (:),  read  '  is  to.'' 
Sign  of  Subtraction  is  — ,  read  'wmwms.' 

Signs  of  Aggregation.  Signs  used  to  group  numbers  that  are  to  be  re- 
garded as  a  single  number. 

They  are  parentheses,  ()  ;  brackets,  []  ;  braces,  {} ;  the  vinculum,  ; 
and  the  vertical  bar,  \ . 

Signs  of  Direction.     Same  as  Signs  of  Quality. 

Signs  of  Opposition.     Same  as  Signs  of  Quality. 

Signs  of  Quality.    The  signs  +  and  —  when  used  to  denote  positive 
and  negative  numbers. 
Similar  Fractions.    Fractions  that  have  the  same  denominator. 

Similar  Radicals.  Radicals  that  in  their  simplest  form  are  of  the 
same  order  and  have  the  same  radicand. 

Similar  Terms.  Terms  that  contain  the  same  letters  with  the  same 
exponents. 

Simple  Equation.  An  integral  equation  that  involves  only  the  first 
power  of  one  unknown  number  in  any  term  when  similar  terms  have  been 
united. 

Simple  Expression.     Same  as  Monomial. 

Simplest  Form  of  a  Radical.  A  radical  is  in  its  simplest  foi-m  when 
the  index  of  the  root  is  as  small  as  possible,  apd  when  the  radicand  is  in- 
tegral and  contains  no  factor  that  is  a  perfect  power  whose  exponent  cor- 
responds with  the  index  of  the  root. 

Simultaneous  Equations.  Two  or  more  equations  that  are  satisfied  by 
the  same  set  or  sets  of  values  of  the  unknown  numbers  form  a  system  of 
simultaneous  equations. 

Solving  an  Equation.    Finding  the  roots  of  an  equation. 

Square.     Same  as  Second  Power. 


320  GLOSSARY 

Square  Root.     One  of  the  two  equal  factors  of  a  number. 

Substitution.     When  a  particular  number  takes  the  place  of  a  letter, 
or  general  number,  the  process  is  called  substitution. 

Subtraction.     The  process  of  finding  one  of  two  numbers  when  their 
sum  and  the  other  number  are  given. 
Subtraction  is  the  inverse  of  addition. 

Subtrahend.    In  subtraction,  the  number  that  is  subtracted. 

Sum.     See  Algebraic  Sum. 

Surd.     The  indicated  root  of  a  rational  number  that  cannot  be  ob- 
tained exactly. 

Symmetrical  Equation.     An  equation  that  is  not  affected  by  interchang- 
ing the  unknown  numbers  involved. 

Term.     An  algebraic  expression  whose  parts  are  not  separated  by  the 
signs  +  or  — . 

Terms  of  a  Fraction.     The  numerator  and  denominator  of  a  fraction. 

Third  Power.     When  a  number  is  used  three  times  as  a  factor,  the 
product  is  called  the  third  power  of  the  number. 

Third  Proportional.     The  consequent  of  the  second  ratio  when  the 
means  of  a  proportion  are  identical. 

Third  Root.    Same  as  Cube  Boot. 

Transposition.     The  process  of  removing  a  term  from  one  member  of 
an  equation  to  the  other. 

Trinomial.     An  algebraic  expression  of  three  terms. 

Trinomial  Square.     A  trinomial  that  is  a  perfect  square. 

Unknown  Number.     A  number  whose  value  is  to  be  found. 

Unlike  Terms.     Same  as  Dissimilar  Terms. 

Whole  Number.     A  unit  or  an  aggregate  of  units. 

X-axis.     The  horizontal  axis  of  reference  is  usually  called  the  x-axis. 

Y-axis.     The  vertical  axis  of  reference  is  usually  called  the  y-axis. 


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